SERIES DE POTENCIAS Y SERIES DE TAYLOR TEORIA

gpereira9 22 views 100 slides Sep 13, 2025
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About This Presentation

Teroria matematica d elas series deTaylor


Slide Content

SERIES DE POTENCIAS Y SERIES DE TAYLOR.
Mallerly Carolina Henriquez Batista.
Juan Toribio Milane, Samuel Salomón.
[email protected]
(November 22, 2023)
Juan T. Milane, [email protected] Ca´lculo Integral. UASD 1 / 1

�
10
�
,

&#3627408475;<0

(−1)
&#3627408475;
(&#3627408485;−2)
&#3627408475;
(&#3627408475;+1)
2

&#3627408475;<0
, lim
??????→0
arcsen (&#3627408485;)
&#3627408485;

Índice
Objetivos
Palabras
Claves Temas Principales
Series de Potencia
Series de Taylor y
Maclaurin
Conclusión
Recursos
Retroalimentación
Interrogantes
Ejercicios
Propuestos
Frase
Motivacional
Bibliografía

Objetivo General
Explicar todo lo relacionado a las series de
Potencias y Series de Taylor, así como
resolver ejercicios afines.

Objetivos Específicos
•Aprender a calcular el intervalo de convergencia de una serie de
potencia.
•Realizar operaciones con series de potencias
•Encontrar las series de potencias que definen una función dada
•Comprender la utilidad de las Series de Taylor y Maclaurin
•Utilizar los softwares de symbolab y WolframAlpha de manera correcta
para calcular los ejercicios.

Retroalimentación
de
Temas

¿Cómo utilizar Symbolab y WolframAlpha?

Límite de una Sucesión
Lim
&#3627408475;→∞
&#3627408462;
&#3627408475;=&#3627408447;
Existe No existe
Converge Diverge
&#3627408462;
&#3627408475;= [
&#3627408475;
&#3627408475;+1
]

1
2
,
2
3
,
3
4
,
4
5
,…
lim
&#3627408475;→∞
[
&#3627408475;
&#3627408475;+1
]=1
&#3627408475;
&#3627408475;+1
&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408462; 1
Ejemplo

Series Infinitas
&#3627408462;
&#3627408475;=&#3627408462;
1+&#3627408462;
2+&#3627408462;
3+…+&#3627408462;
&#3627408475;+⋯

&#3627408475;<1

Términos de
la Serie
[&#3627408462;
&#3627408475;]→&#3627408454;&#3627408482;&#3627408464;&#3627408466;&#3627408480;&#3627408470;ó&#3627408475;
&#3627408454;
&#3627408475;=&#3627408462;
1+&#3627408462;
2+&#3627408462;
3 →&#3627408454;&#3627408482;&#3627408464;&#3627408466;&#3627408480;&#3627408470;ó&#3627408475; &#3627408465;&#3627408466; &#3627408454;&#3627408482;&#3627408474;&#3627408462; &#3627408477;&#3627408462;&#3627408479;&#3627408464;&#3627408470;&#3627408462;&#3627408473;&#3627408466;&#3627408480; &#3627408476; &#3627408454;&#3627408466;&#3627408479;&#3627408470;&#3627408466; ??????&#3627408475;&#3627408467;&#3627408470;&#3627408475;&#3627408470;&#3627408481;&#3627408462;
Convergencia y Divergencia
de Series
&#3627408454;
&#3627408475;=&#3627408462;
1+&#3627408462;
2+&#3627408462;
3+⋯+&#3627408462;
&#3627408475;
[&#3627408462;
&#3627408475;]

&#3627408475;<1

&#3627408454;
&#3627408475;&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408462; &#3627408454;→ [&#3627408462;
&#3627408475;]

&#3627408475;<1
&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;
Suma de la serie
&#3627408454;=&#3627408462;
1+&#3627408462;
2+&#3627408462;
3+⋯+&#3627408462;
&#3627408475;+⋯; &#3627408454;= [&#3627408462;
&#3627408475;]

&#3627408475;<1

&#3627408454;
&#3627408475; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;→ [&#3627408462;
&#3627408475;]

&#3627408475;<1
&#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;
[ ]

&#3627408475;<1

Índice, Variable o Límite Inferior
Argumento
Límite Superior

Teorema del Valor
Absoluto
&#3627408462;
&#3627408475;
lim
&#3627408475;→∞
|&#3627408462;
&#3627408475;| =0
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;=0
Entonces
Convergencia Absoluta y Condicional
|&#3627408462;
&#3627408475;|→Converge &#3627408462;
&#3627408475;→Converge &#3627408466;&#3627408475;&#3627408481;&#3627408476;&#3627408475;&#3627408464;&#3627408466;&#3627408480;
&#3627408462;
&#3627408475;→Absolutamente Convergente
&#3627408462;
&#3627408475;→Converge |&#3627408462;
&#3627408475;|→Diverge
&#3627408462;
&#3627408475;→Condicionalmente Convergente

Series p y Series Armónicas

1
&#3627408475;
&#3627408477;
=
1
1
&#3627408477;
+
1
2
&#3627408477;
+
1
3
&#3627408477;
+⋯

&#3627408475;<1
→&#3627408402;&#3627408414;&#3627408427;&#3627408418;&#3627408414; &#3627408425;

1
&#3627408475;
=1+
1
2
+
1
3
+⋯

&#3627408475;<1
→&#3627408402;&#3627408414;&#3627408427;&#3627408418;&#3627408414; &#3627408410;&#3627408427;&#3627408422;ó&#3627408423;&#3627408418;&#3627408412;&#3627408410;
&#3627408529;=&#3627409359;
&#3627408386;&#3627408424;&#3627408423;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414;&#3627408423;&#3627408412;&#3627408418;&#3627408410; &#3627408413;&#3627408414; &#3627408402;&#3627408414;&#3627408427;&#3627408418;&#3627408414;&#3627408428; &#3627408425;

1
&#3627408475;
&#3627408477;
=
1
1
&#3627408477;
+
1
2
&#3627408477;
+
1
3
&#3627408477;
+⋯

&#3627408475;<1

&#3627409359;.&#3627408386;&#3627408424;&#3627408423;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414; &#3627408428;&#3627408418; &#3627408425;>&#3627409359;
&#3627409360;.&#3627408387;&#3627408418;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414; &#3627408428;&#3627408418; &#3627409358;<&#3627408425;≤&#3627409359;
&#3627408389;&#3627408430;&#3627408423;&#3627408412;&#3627408418;ó&#3627408423; &#3627408435;&#3627408414;&#3627408429;&#3627408410; &#3627408413;&#3627408414; &#3627408401;&#3627408418;&#3627408414;&#3627408422;&#3627408410;&#3627408423;&#3627408423;
??????&#3627408485;=
1
&#3627408475;
??????

&#3627408475;<1

−&#3627408425;

1
&#3627408475;
;&#3627408477;

&#3627408475;<1

→Criterio del Término n−símo para la divergencia
&#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;≠0→ &#3627408462;
&#3627408475; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;

&#3627408475;<1

&#3627408475;
&#3627408477;

&#3627408475;<1

&#3627408425;=&#3627409358;

1
&#3627408475;
0
= 1

&#3627408475;<1

&#3627408475;<1

lim
&#3627408475;→∞
&#3627408475;
&#3627408477;
=∞
lim
&#3627408475;→∞
1=1
1

&#3627408475;<1 &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;

Series Alternadas o Alternantes
(Criterio de Leibniz)
&#3627408454;&#3627408466;&#3627408462; &#3627408462;
&#3627408475;>0
(−1)
&#3627408475;
&#3627408462;
&#3627408475;

&#3627408475;<1
(−1)
&#3627408475;:1
&#3627408462;
&#3627408475;

&#3627408475;<1

&#3627409359;.lim
&#3627408475;→∞
&#3627408462;
&#3627408475;=0
2. &#3627408462;
&#3627408475;:1≤&#3627408462;
&#3627408475;,para todo &#3627408475;
Convergen si:
(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1

Ejemplo
Determinar si la siguiente serie es divergente o convergente
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;→lim
&#3627408475;→∞
1
&#3627408475;
=0
&#3627408462;
&#3627408475;=
1
&#3627408475;

&#3627408462;
&#3627408475;:1≤&#3627408462;
&#3627408475;→
1
&#3627408475;+1

1
&#3627408475;

&#3627408462;
&#3627408475;:1=
1
&#3627408475;+1

(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1
es convergente
¿&#3627408440;&#3627408480; &#3627408462;&#3627408463;&#3627408480;&#3627408476;&#3627408473;&#3627408481;&#3627408462;&#3627408474;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408476;
&#3627408464;&#3627408476;&#3627408475;&#3627408465;&#3627408470;&#3627408464;&#3627408470;&#3627408476;&#3627408475;&#3627408462;&#3627408473;&#3627408474;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466;?
&#3627408462;
&#3627408475;=(−1)
&#3627408475;:1
1
&#3627408475;

(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1

(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1

(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1
=
(−1)
&#3627408475;:1
&#3627408475;

&#3627408475;<1

=
(−1)
&#3627408475;:1
&#3627408475;

&#3627408475;<1

=
1
|&#3627408475;|

&#3627408475;<1

Determinar la
convergencia de la serie
(−1)
&#3627408475;:1
1
&#3627408475;
&#3627408466;&#3627408480; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466;

&#3627408475;<1

(−1)
&#3627408475;:1
1
&#3627408475;

&#3627408475;<1
es condicionalmente convergente

Series Geométricas
&#3627408462;&#3627408479;
&#3627408475;
=&#3627408462;+&#3627408462;&#3627408479;+&#3627408462;&#3627408479;
2
+⋯

&#3627408475;<0
+&#3627408462;&#3627408479;
&#3627408475;
+⋯,&#3627408462;≠0
Convergencia de una Serie Geométrica
Diverge si &#3627408479;≥1. &#3627408454;&#3627408470; 0<&#3627408479;<1,→&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408462; &#3627408473;&#3627408462; &#3627408480;&#3627408482;&#3627408474;&#3627408462;:
r es la razón
&#3627408462;&#3627408479;
&#3627408475;
=
&#3627408462;
1−&#3627408479;
, 0<&#3627408479;<1

&#3627408475;<0


2
5
&#3627408475;

&#3627408475;<0


2
5
&#3627408475;

&#3627408475;<0
= 2(
1
5
)
&#3627408475;

&#3627408475;<0

&#3627408462;=2
&#3627408479;
&#3627408475;
=(
1
5
)
&#3627408475;

=21+
2
5
+2
1
10
+⋯
??????&#3627408476;&#3627408474;&#3627408476; 0<
1
5
<1,&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;,&#3627408477;&#3627408476;&#3627408479; &#3627408473;&#3627408476; &#3627408478;&#3627408482;&#3627408466;:
&#3627408454;=
2
1−(
1
5
)

&#3627408454;=
5
2

Ejemplo

&#3627408462;&#3627408479;
&#3627408475;
=&#3627408462;+&#3627408462;&#3627408479;+&#3627408462;&#3627408479;
2
+⋯

&#3627408475;<0
+&#3627408462;&#3627408479;
&#3627408475;
+⋯,&#3627408462;≠0
&#3627408531;=&#3627409358;
&#3627408462;(0)
&#3627408475;
=&#3627408462;(0)
0
+&#3627408462;0
1
+&#3627408462;0
2
+⋯

&#3627408475;<0
+&#3627408462;0
&#3627408475;
+⋯,&#3627408462;≠0
=&#3627408462;1+0+0+⋯
&#3627408462;(0)
&#3627408475;
=&#3627408462;

&#3627408475;<0

&#3627408531;=&#3627409359;
&#3627408462;(1)
&#3627408475;
=&#3627408462;(1)
0
+&#3627408462;(1)
1
+&#3627408462;(1)
2
+⋯

&#3627408475;<0
+&#3627408462;(1)
&#3627408475;
+⋯,&#3627408462;≠0
=&#3627408462;1+&#3627408462;(1)+&#3627408462;(1)+⋯
&#3627408462;(1)
&#3627408475;
=&#3627408462;+&#3627408462;+&#3627408462;+⋯

&#3627408475;<0

Criterio del Término n−símo para la divergencia
&#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;≠0→ &#3627408462;
&#3627408475; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;

&#3627408475;<1

&#3627408462;(1)
&#3627408475;
&#3627408466;&#3627408480; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408477;&#3627408462;&#3627408479;&#3627408462; &#3627408462;≠0

&#3627408475;<0

&#3627408462;(0)
&#3627408475;
&#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;

&#3627408475;<0

Criterio de la
Integral
Criterios de Convergencia
&#3627408459;≥1 &#3627408486; &#3627408462;
&#3627408475;=&#3627408467;(&#3627408475;)
[&#3627408462;
&#3627408475;]

&#3627408475;<1
&#3627408486; &#3627408467;(&#3627408485;)

1

Ambas Convergen o
Ambas Divergen
Criterio del
Cociente
&#3627408462;
&#3627408475;≠0
1. &#3627408462;
&#3627408475; &#3627408466;&#3627408480; &#3627408462;&#3627408463;&#3627408480;&#3627408476;&#3627408473;&#3627408482;&#3627408481;&#3627408462;&#3627408474;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408480;&#3627408470;
lim
&#3627408475;→∞
&#3627408462;??????+1
&#3627408462;
??????
<1
2. &#3627408462;
&#3627408475; &#3627408466;&#3627408480; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
>1 ó
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
=∞
3.&#3627408440;&#3627408473; &#3627408464;&#3627408479;&#3627408470;&#3627408481;&#3627408466;&#3627408479;&#3627408470;&#3627408476; &#3627408465;&#3627408466;&#3627408473; &#3627408464;&#3627408476;&#3627408464;&#3627408470;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408475;&#3627408476; &#3627408466;&#3627408480; &#3627408464;&#3627408476;&#3627408475;&#3627408464;&#3627408473;&#3627408482;&#3627408486;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
&#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
=1
Criterio de la
Raíz
&#3627408462;
&#3627408475;
1. &#3627408462;
&#3627408475; &#3627408466;&#3627408480; &#3627408462;&#3627408463;&#3627408480;&#3627408476;&#3627408473;&#3627408482;&#3627408481;&#3627408462;&#3627408474;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466; &#3627408480;&#3627408470;
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
<1
2. &#3627408462;
&#3627408475; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
>1 ó
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
=∞
3.&#3627408440;&#3627408473; &#3627408464;&#3627408479;&#3627408470;&#3627408481;&#3627408466;&#3627408479;&#3627408470;&#3627408476; &#3627408465;&#3627408466; &#3627408473;&#3627408462; &#3627408479;&#3627408462;í&#3627408487; &#3627408475;&#3627408476; &#3627408466;&#3627408480; &#3627408464;&#3627408476;&#3627408475;&#3627408464;&#3627408473;&#3627408482;&#3627408486;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
&#3627408480;&#3627408470;lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
=1

Polinomios de Taylor y Maclaurin
??????
&#3627408475;&#3627408485;=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;+
??????´´(&#3627408464;)
2!
+⋯+
??????
??????
&#3627408464;
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
→Polinomio de Taylor de grado n para f
Para c=0→P
&#3627408475;&#3627408485;=&#3627408467;0+&#3627408467;´0+
??????´´(0)
2!
&#3627408485;
2
+
??????´´´(0)
3!
&#3627408485;
3
+⋯+
??????
??????
(0)
&#3627408475;!
&#3627408485;
&#3627408475;
→Polinomio de Maclaurin
Ejemplo
Encontrar los polinomios de Taylor ??????
0,??????
1 &#3627408486;??????
2 para f(x)= Ln(x) centrada en c=1
&#3627408467;&#3627408485;=ln&#3627408485; →&#3627408467;1=ln1=0
&#3627408467;´&#3627408485;=
1
&#3627408485;
→&#3627408467;´1=
1
1
=1
&#3627408467;´´&#3627408485;=−
1
&#3627408485;
2
→&#3627408467;´´1=−
1
1
2
=−1
??????
0=&#3627408467;1=0
??????
1=&#3627408467;1+&#3627408467;´1&#3627408485;−1=&#3627408485;−1
??????
2=&#3627408467;1+&#3627408467;´1&#3627408485;−1+
&#3627408467;´´(1)
2!
(&#3627408485;−1)
2
=&#3627408485;−1−
1
2
(&#3627408485;−1)
2

Residuo de un Polinomio de Taylor
&#3627408467;&#3627408485;=??????
&#3627408475;&#3627408485;+&#3627408453;
&#3627408475;(&#3627408485;)
Valor Exacto Valor
Aproximado
Residuo
Así,&#3627408453;
&#3627408475;&#3627408485;=&#3627408467;&#3627408485;−??????
&#3627408475;&#3627408485;.
Error=&#3627408453;
&#3627408475;(&#3627408485;)=&#3627408467;(&#3627408485;)−??????
&#3627408475;(&#3627408485;)
Teorema de Taylor
&#3627408467;&#3627408485;=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−1+
&#3627408467;´´&#3627408464;
2!
(&#3627408485;−&#3627408464;)
2
+⋯+
&#3627408467;
&#3627408475;
&#3627408464;
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
+&#3627408453;
&#3627408475;(&#3627408475;)
&#3627408453;
&#3627408475;&#3627408475;=
&#3627408467;
&#3627408475;:1
&#3627408487;
(&#3627408475;+1)!
(&#3627408485;−&#3627408464;)
&#3627408475;:1

Interrogantes
¿Qué son las Series de Potencias?
¿Cuándo una serie converge o diverge?
¿Cómo se calculan el Radio e Intervalo de Convergencia?
¿Es posible transformar una función en una serie de potencias?
¿Qué son las Series de Taylor y Maclaurin?
¿Cuándo Convergen las series de Taylor y Maclaurin?

Palabras Claves
Series
Convergente
Diverge
Intervalo de Convergencia
Radio de Convergencia

Definición
&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
=&#3627408462;
0+&#3627408462;
1&#3627408485;+&#3627408462;
2&#3627408485;
2
+&#3627408462;
3&#3627408485;
3
+⋯+&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
+⋯

&#3627408475;<0

Coeficientes
Variable
Sucesión de sumas
parciales
&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
=&#3627408462;
0+&#3627408462;
1(&#3627408485;−&#3627408464;)+&#3627408462;
2(&#3627408485;−&#3627408464;)
2
+&#3627408462;
3(&#3627408485;−&#3627408464;)
3
+⋯+&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
+⋯

&#3627408475;<0

c es una constante y es el centro
de la serie ∀ &#3627408485; ∈|&#3627408485;−&#3627408464;|<&#3627408453;

Ejemplos
3. Serie de potencia centrada en 0

&#3627408485;
&#3627408475;
&#3627408475;!

&#3627408475;<0
=1+&#3627408485;+
&#3627408485;
2
2
+
&#3627408485;
3
3!
+⋯
1. Serie de potencia centrada en 1

1
&#3627408475;
(&#3627408485;−1)
&#3627408475;

&#3627408475;<1
=(&#3627408485;−1)+
1
2
(&#3627408485;−1)
2
+
1
3
(&#3627408485;−1)
3
+⋯
2. Serie de potencia centrada en -1
(−1)
&#3627408475;
(&#3627408485;+1)
&#3627408475;

&#3627408475;<0
=1−(&#3627408485;+1)+ (&#3627408485;+1)
2
− (&#3627408485;−1)
3
+⋯

Radio e Intervalo de Convergencia
¿Qué es el Radio de Convergencia? ¿Qué es el Intervalo de Convergencia?
c c c

Convergencia de una Serie de Potencias
1. La serie converge sólo en c
&#3627408453;=0
2. Existe un número real &#3627408453;>0 tal que la serie converge
absolutamente para &#3627408485;−&#3627408464;<&#3627408453;, y diverge para
&#3627408485;−&#3627408464;>&#3627408453;
3. La serie converge absolutamente
para todo x
&#3627408453;=∞

¿Cómo determinar el Intervalo y el Radio de
Convergencia?
Radio de Convergencia
1.Cuando la serie converge sólo a c→&#3627408453;=0
2.Cuando la serie converge para todo x→&#3627408453;=∞
3.¿?
Ejemplo

(&#3627408485;+1)
&#3627408475;
&#3627408475;2
&#3627408475;

&#3627408475;<0
1.lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
<1
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
(&#3627408537;+&#3627409359;)
&#3627408527;:&#3627409359;
&#3627408527;+&#3627409359;&#3627409360;
&#3627408527;:&#3627409359;
(&#3627408537;+&#3627409359;)
&#3627408527;
&#3627408527;&#3627409360;
&#3627408527;
<&#3627409359;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
(&#3627408537;+&#3627409359;)
&#3627408527;
(&#3627408537;+&#3627409359;)
&#3627409359;
&#3627408527;&#3627409360;
&#3627408527;
&#3627408527;+&#3627409359;&#3627409360;
&#3627408527;
&#3627409360;
&#3627409359;
(&#3627408537;+&#3627409359;)
&#3627408527;
<&#3627409359;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
(&#3627408537;+&#3627409359;)
&#3627409359;
&#3627408527;
(&#3627408527;+&#3627409359;)&#3627409360;
&#3627409359;
<&#3627409359;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
&#3627408537;+&#3627409359;
&#3627409360;
&#3627408527;
&#3627408527;+&#3627409359;
<&#3627409359;
&#3627408537;+&#3627409359;
&#3627409360;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
&#3627408527;
&#3627408527;+&#3627409359;
<&#3627409359;
&#3627408537;+&#3627409359;
&#3627409360;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
1
1+
1
&#3627408475;
<1
&#3627408537;+&#3627409359;
&#3627409360;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
&#3627409359;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
&#3627409359;−
&#3627409359;
&#3627408527;
<&#3627409359;
&#3627408537;+&#3627409359;
&#3627409360;
(&#3627409359;)<&#3627409359;
&#3627408537;+&#3627409359;
&#3627409360;
<&#3627409359;
&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
(&#3627408537;+&#3627409359;)
&#3627408527;:&#3627409359;
&#3627408527;&#3627409360;
&#3627408527;
&#3627408527;+&#3627409359;&#3627409360;
&#3627408527;:&#3627409359;
(&#3627408537;+&#3627409359;)
&#3627408527;
<&#3627409359;
|&#3627408537;+&#3627409359;|
&#3627409360;
lim
&#3627408527;→∞
&#3627408527;
&#3627408527;
&#3627408527;
&#3627408527;
+
&#3627409359;
&#3627408527;
<&#3627409359;

(&#3627408485;+1)
&#3627408475;
&#3627408475;2
&#3627408475;

&#3627408475;<0
Converge

¿Cómo podemos determinar el radio y el intervalo de convergencia
para una serie de potencias, aplicando el criterio de la razón de la serie
absoluta?
&#3627408463;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;

&#3627408475;<0

&#3627408463;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;

&#3627408475;<0
= &#3627408462;
&#3627408475;

&#3627408475;<0

lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
<1
&#3627408485;−&#3627408464;lim
&#3627408475;→∞
&#3627408463;
&#3627408475;:1
&#3627408463;
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408463;
&#3627408475;:1(&#3627408485;−&#3627408464;)
&#3627408475;:1
&#3627408463;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
<1
&#3627408454;&#3627408470;:&#3627408485;=&#3627408464;
0<1
&#3627408454;&#3627408470;:&#3627408485;≠&#3627408464;
&#3627408485;−&#3627408464;(lim
&#3627408475;→∞
&#3627408463;
&#3627408475;:1
&#3627408463;
&#3627408475;
)<1 ,
1
&#3627408453;
=lim
&#3627408475;→∞
&#3627408463;
&#3627408475;:1
&#3627408463;
&#3627408475;

|&#3627408485;−&#3627408464;|
1
&#3627408453;
<1
|&#3627408537;−&#3627408516;|<&#3627408505;
−&#3627408453;<&#3627408485;−&#3627408464;<&#3627408453;
c−&#3627408505;<&#3627408537;<&#3627408516;+&#3627408505; &#3627408485;∈(&#3627408464;−&#3627408453;,&#3627408464;+&#3627408453;)
&#3627408453;=lim
&#3627408475;→∞
&#3627408463;
&#3627408475;
&#3627408463;
&#3627408475;:1

&#3627408537;+&#3627409359;
&#3627409360;
<&#3627409359;
&#3627408537;+&#3627409359;<&#3627409360; ←&#3627408401;=&#3627409360;
−&#3627409360;<&#3627408537;+&#3627409359;<&#3627409360;
−&#3627409359;−&#3627409360;<&#3627408537;+&#3627409359;−&#3627409359;<&#3627409360;−&#3627409359;
−&#3627409361;<&#3627408537;<&#3627409359;
&#3627408392;&#3627408423;&#3627408429;&#3627408414;&#3627408427;&#3627408431;&#3627408410;&#3627408421;&#3627408424; &#3627408413;&#3627408414; &#3627408386;&#3627408424;&#3627408423;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414;&#3627408423;&#3627408412;&#3627408418;&#3627408410;
(−&#3627409361;,&#3627409359;)
&#3627408463;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
converge

&#3627408475;<0

Ejemplo 2

(&#3627408485;−5)
&#3627408475;
7
&#3627408475;

&#3627408475;<0

&#3627408421;&#3627408418;&#3627408422;
&#3627408527;→∞
??????
&#3627408527;
&#3627408527;
<&#3627409359;

(&#3627408485;−5)
&#3627408475;
7
&#3627408475;
=lim
&#3627408475;→∞
(&#3627408485;−5)
&#3627408475;
7
&#3627408475;
<1
??????

&#3627408475;<0

lim
&#3627408475;→∞
(&#3627408485;−5)
&#3627408475;
7
&#3627408475;
1
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;−5
&#3627408475;
7
&#3627408475;
1
&#3627408475;

<1
lim
&#3627408475;→∞
(&#3627408485;−5)
&#3627408475;
1
&#3627408475;
(7
&#3627408475;
)
1
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;−5
7
<1
lim
&#3627408475;→∞
(&#3627408485;−5)
&#3627408475;
7
&#3627408475;
??????
<1
=
&#3627408485;−5
7
<1
&#3627408485;−5
7
<1
&#3627408485;−5<7
&#3627408505;=&#3627409365;
−7<&#3627408485;−5<7
5−7<&#3627408485;<7+5
−2<&#3627408485;<12
&#3627408392;&#3627408423;&#3627408429;&#3627408414;&#3627408427;&#3627408431;&#3627408410;&#3627408421;&#3627408424; &#3627408413;&#3627408414; &#3627408386;&#3627408424;&#3627408423;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414;&#3627408423;&#3627408412;&#3627408418;&#3627408410;
(−&#3627409360;,&#3627409359;&#3627409360;)

Convergencia en los puntos terminales
1.Sea R=0→Intervalo de convergencia=[c]
2.Sea R=∞→Intervalo de convergencia=(−∞,∞)
3.Sea Radio=&#3627408453;→Intervalo de convergencia=(c−R,c+R)
3.1.(&#3627408464;−&#3627408453;,&#3627408464;+&#3627408453;)
3.2.(&#3627408464;−&#3627408453;,&#3627408464;+&#3627408453;]
3.3.[&#3627408464;−&#3627408453;,&#3627408464;+&#3627408453;)
3.4.[&#3627408464;−&#3627408453;,&#3627408464;+&#3627408453;]

Derivación e Integración de Series de Potencias
&#3627408467;&#3627408485;= &#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;

&#3627408475;<0

=&#3627408462;
0+&#3627408462;
1(&#3627408485;−&#3627408464;)+&#3627408462;
2(&#3627408485;−&#3627408464;)
2
+&#3627408462;
3(&#3627408485;−&#3627408464;)
3
+⋯
&#3627408453;>0
Intervalo(c−R,c+R)
1.&#3627408467;´&#3627408485;= &#3627408475;&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;;1∞
&#3627408475;<1
=&#3627408462;
1+2&#3627408462;
2&#3627408485;−&#3627408464;+3&#3627408462;
3&#3627408485;−&#3627408464;
2
+⋯
2. &#3627408467;&#3627408485;&#3627408465;&#3627408485;=??????+ &#3627408462;
&#3627408475;
(&#3627408485;−&#3627408464;)
&#3627408475;:1
&#3627408475;+1

&#3627408475;<0

=??????+&#3627408462;
0&#3627408485;−&#3627408464;+&#3627408462;
1
&#3627408485;−&#3627408464;
2
2
+&#3627408462;
2
&#3627408485;−&#3627408464;
3
3
+⋯
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;
&#3627408475;
=&#3627408475;&#3627408485;
&#3627408475;;1
,
&#3627408485;
&#3627408475;
&#3627408465;&#3627408485;=
&#3627408485;
&#3627408475;:1
&#3627408475;+1
+??????,&#3627408477;&#3627408462;&#3627408479;&#3627408462; &#3627408475;≠−1

Ejemplo
Dada la función: &#3627408467;&#3627408485;=
&#3627408485;
&#3627408475;
&#3627408475;
=&#3627408485;+
&#3627408485;
2
2
+
&#3627408485;
3
3

&#3627408475;<1
+⋯
Calcular los intervalos de convergencia para cada una de las siguientes expresiones:
&#3627408462;) &#3627408467;&#3627408485;&#3627408465;&#3627408485;, &#3627408463;)&#3627408467;&#3627408485;, &#3627408464;)&#3627408467;´(&#3627408485;)
&#3627408462;) &#3627408467;&#3627408485;&#3627408465;&#3627408485;=??????+
&#3627408485;
&#3627408475;:1
&#3627408475;(&#3627408475;+1)

&#3627408475;<1

=??????+
&#3627408485;
2
1(2)
+
&#3627408485;
3
2(3)
+
&#3627408485;
4
3(4)
+⋯
&#3627408467;´&#3627408485;= &#3627408475;&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;;1

&#3627408475;<1

=&#3627408462;
1+2&#3627408462;
2&#3627408485;−&#3627408464;+3&#3627408462;
3&#3627408485;−&#3627408464;
2
+⋯
&#3627408467;&#3627408485;&#3627408465;&#3627408485;=??????+ &#3627408462;
&#3627408475;
(&#3627408485;−&#3627408464;)
&#3627408475;:1
&#3627408475;+1

&#3627408475;<0

=??????+&#3627408462;
0&#3627408485;−&#3627408464;+&#3627408462;
1
&#3627408485;−&#3627408464;
2
2
+&#3627408462;
2
&#3627408485;−&#3627408464;
3
3
+⋯
&#3627408464;)&#3627408467;´&#3627408485;= &#3627408485;
&#3627408475;;1

&#3627408475;<1

=1+&#3627408485;+&#3627408485;
2
+&#3627408485;
3
+⋯
Paso 1

&#3627408462;)Para &#3627408467;&#3627408485;&#3627408465;&#3627408485;,la serie
&#3627408485;
&#3627408475;:1
&#3627408475;(&#3627408475;+1)

&#3627408475;<1

→Intervalo de Convergencia:[−1,1]
&#3627408463;)Para fx,la serie
&#3627408485;
&#3627408475;
&#3627408475;

&#3627408475;<1

→Intervalo de Convergencia:[−1,1)
&#3627408464;)Para f´x,la serie &#3627408485;
&#3627408475;;1

&#3627408475;<1

→Intervalo de Convergencia:(−1,1)
Paso 2: Utilizar uno de los criterios para determinar el radio e intervalo de
convergencia
&#3627408453;=1
Intervalo de Convergencia(−1,1)

¿Cómo representar una función en una Serie
Geométrica de Potencias?
Serie Geométrica
&#3627408462;&#3627408479;
&#3627408475;
=&#3627408462;+&#3627408462;&#3627408479;+&#3627408462;&#3627408479;
2
+&#3627408462;&#3627408479;
3
+⋯=
&#3627408462;
1−&#3627408479;

&#3627408475;<0

&#3627408479;<1
&#3627408462;
&#3627408475;=1
&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
=&#3627408462;
0+&#3627408462;
1(&#3627408485;−&#3627408464;)+&#3627408462;
2(&#3627408485;−&#3627408464;)
2
+&#3627408462;
3(&#3627408485;−&#3627408464;)
3
+⋯+&#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;
+⋯

&#3627408475;<0

&#3627408464;=0
&#3627408485;
&#3627408475;
=&#3627408485;
0
+&#3627408485;
1
+&#3627408485;
2
+&#3627408485;
3
+⋯

&#3627408475;<0
←Converge en
1
1−&#3627408485;

&#3627408485;<1
−1<&#3627408485;<1
(1)&#3627408485;
&#3627408475;
=&#3627408485;
0
+&#3627408485;
1
+&#3627408485;
2
+&#3627408485;
3
+⋯

&#3627408475;<0

−1<&#3627408479;<1 &#3627408462;&#3627408479;
&#3627408475;
=
&#3627408462;
1−&#3627408479;
→&#3627408467;&#3627408485;=
&#3627408462;
1−&#3627408485;

&#3627408475;<0

&#3627408467;&#3627408485;=
2
&#3627408485;+4

Ejemplo
Determina el intervalo de convergencia de la siguiente función
&#3627408462;&#3627408479;
&#3627408475;
=
&#3627408462;
1−&#3627408479;
→&#3627408467;&#3627408485;=
&#3627408462;
1−&#3627408485;

&#3627408475;<0

2
&#3627408485;+4
=
2
4+&#3627408485;

=
2
4−(−&#3627408485;)

=
2
4
4
4
−−
&#3627408485;
4

=
1
2
1−−
&#3627408485;
4

&#3627408462;=
1
2

&#3627408479;=−
&#3627408485;
4


1
2

&#3627408485;
4
&#3627408475;

&#3627408475;<0

= −
1
2
&#3627408475;
&#3627408485;
4
&#3627408475;

&#3627408475;<0

&#3627408479;<1 →
&#3627408485;
4
<1
&#3627408485;<4 &#3627408453;=4
−4<&#3627408485;<4
Intervalo de convergencia fx=
2
&#3627408485;+4

(−4,4)

Operaciones Elementales con Series de Potencias
Suma-Resta-Multiplicación-División
&#3627408454;&#3627408466;&#3627408462; &#3627408467;&#3627408485;= &#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
&#3627408486; &#3627408468;&#3627408485;= &#3627408463;
&#3627408475;&#3627408485;
&#3627408475;

1.&#3627408467;&#3627408472;&#3627408485;= &#3627408462;
&#3627408475;&#3627408472;
&#3627408475;
&#3627408485;
&#3627408475;

&#3627408475;<0

2.&#3627408467;&#3627408485;
??????
= &#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
??????

&#3627408475;<0

3.&#3627408467;&#3627408485;±&#3627408468;&#3627408485;= (&#3627408462;
&#3627408475;±&#3627408463;
&#3627408475;) &#3627408485;
&#3627408475;

&#3627408475;<0

&#3627408388;&#3627408419;&#3627408414;&#3627408422;&#3627408425;&#3627408421;&#3627408424;:&#3627408402;&#3627408430;&#3627408422;&#3627408410;&#3627408427; &#3627408421;&#3627408410;&#3627408428; &#3627408428;&#3627408418;??????&#3627408430;&#3627408418;&#3627408414;&#3627408423;&#3627408429;&#3627408414; &#3627408425;&#3627408424;&#3627408429;&#3627408414;&#3627408423;&#3627408412;&#3627408418;&#3627408410;&#3627408428;
a) &#3627408485;
&#3627408475;
&#3627408486; &#3627408463;)
&#3627408485;
2
&#3627408475;

&#3627408475;<0

&#3627408475;<0

(&#3627408485;
&#3627408475;
)+
&#3627408485;
&#3627408475;
2
&#3627408475;


&#3627408475;<0


2
&#3627408475;
&#3627408485;
&#3627408475;
+&#3627408485;
&#3627408475;
2
&#3627408475;

&#3627408475;<0


2
&#3627408475;
&#3627408485;
&#3627408475;
2
&#3627408475;
+
&#3627408485;
&#3627408475;
2
&#3627408475;

&#3627408475;<0

&#3627408485;
&#3627408475;
+
&#3627408485;
&#3627408475;
2
&#3627408475;

&#3627408475;<0

1+
1
2
&#3627408475;
&#3627408485;
&#3627408475;

&#3627408475;<0

&#3627408462;) &#3627408485;
&#3627408475;
→??????&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408466;&#3627408475;:(−1,1)

&#3627408475;<0

&#3627408463;)
&#3627408485;
2
&#3627408475;
→??????&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408466;&#3627408475;:(−2,2)

&#3627408475;<0

&#3627408462;+&#3627408463;) 1+
1
2
&#3627408475;
&#3627408485;
&#3627408475;
→??????&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408466;&#3627408475; (−1,1)

&#3627408475;<0

Multiplicación de Series de Potencias
&#3627408462;)
1
5
&#3627408475;
&#3627408475;
&#3627408485;
&#3627408475;

&#3627408475;<1
&#3627408460; &#3627408463;) (&#3627408475;+1)
2
&#3627408485;
&#3627408475;

&#3627408475;<0


1
5
&#3627408475;
&#3627408475;
&#3627408485;
&#3627408475;

&#3627408475;<1
=
1
5
&#3627408485; +
1
50
&#3627408485;
2
+
1
250
&#3627408485;
3
+⋯
(&#3627408475;+1)
2
&#3627408485;
&#3627408475;
=

&#3627408475;<0
1&#3627408485;
0
+4&#3627408485;
1
+9&#3627408485;
2
+16&#3627408485;
3
+⋯
&#3627409359;
&#3627409363;
&#3627408537;
&#3627409359;
+
&#3627409362;
&#3627409363;
&#3627408537;
&#3627409360;
+
&#3627409367;
&#3627409363;
&#3627408537;
&#3627409361;
+⋯
+
&#3627409359;
&#3627409363;&#3627409358;
&#3627408537;
&#3627409360;
+
&#3627409360;
&#3627409360;&#3627409363;
&#3627408537;
&#3627409361;
+⋯
+
&#3627409359;
&#3627409360;&#3627409363;&#3627409358;
&#3627408537;
&#3627409361;
+⋯
1
5
&#3627408485;
1
+
4
5
+
1
50
&#3627408485;
2
+
9
5
+
2
25
+
1
250
&#3627408485;
3
+⋯
1
5
&#3627408485;
1
+
4
5
+
1
50
&#3627408485;
2
+
9
5
+
2
25
+
1
250
&#3627408485;
3
+⋯
&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
&#3627408475;≥0
&#3627408463;
&#3627408475;&#3627408485;
&#3627408475;
&#3627408475;≥0
=
&#3627408475;≥0
&#3627408462;
&#3627408471;&#3627408463;
&#3627408475;;&#3627408471;
&#3627408475;
&#3627408471;≥0
&#3627408485;
&#3627408475;


&#3627409359;
&#3627409363;
&#3627408527;
&#3627408527;
&#3627408537;
&#3627408527;

&#3627408527;<&#3627409359;
&#3627408527;+&#3627409359;
&#3627409360;
&#3627408537;
&#3627408527;

&#3627408527;<&#3627409358;

Orden:3

División de Series de
Potencias

1
&#3627408475;
2
&#3627408485;
&#3627408475;

&#3627408475;<1
&#3627408475;+2&#3627408485;
&#3627408475;

&#3627408475;<0


1
&#3627408475;
2
&#3627408485;
&#3627408475;∞
&#3627408475;<1
&#3627408475;+2&#3627408485;
&#3627408475;∞
&#3627408475;<0
= &#3627408462;
&#3627408475;&#3627408485;
&#3627408475;

&#3627408475;<0

Orden:3

1
&#3627408475;
2
&#3627408485;
&#3627408475;

&#3627408475;<1
= &#3627408475;+2&#3627408485;
&#3627408475;

&#3627408475;<0
&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;

&#3627408475;<0

1&#3627408485;
1
+
1
4
&#3627408485;
2
+
1
9
&#3627408485;
3
+⋯=(2&#3627408485;
0
+3&#3627408485;
1
+4&#3627408485;
2
+5&#3627408485;
3
+⋯)(&#3627408462;
0&#3627408485;
0
+&#3627408462;
1&#3627408485;
1
+&#3627408462;
2&#3627408485;
2
+&#3627408462;
3&#3627408485;
3
+⋯)
2&#3627408485;
0
+3&#3627408485;
1
+4&#3627408485;
2
+5&#3627408485;
3
+⋯
&#3627408462;
0&#3627408485;
0
+&#3627408462;
1&#3627408485;
1
+&#3627408462;
2&#3627408485;
2
+&#3627408462;
3&#3627408485;
3
+⋯
2&#3627408462;
0&#3627408485;
0
+2&#3627408462;
1&#3627408485;
1
+2&#3627408462;
2&#3627408485;
2
+2&#3627408462;
3&#3627408485;
3
+⋯
+3&#3627408462;
0&#3627408485;
1
+3&#3627408462;
1&#3627408485;
2
+3&#3627408462;
2&#3627408485;
3
+⋯
+4&#3627408462;
0&#3627408485;
2
+4&#3627408462;
1&#3627408485;
3
+⋯
+ 5&#3627408462;
0&#3627408485;
3
+⋯
2&#3627408462;
0&#3627408485;
0
+3&#3627408462;
0+2&#3627408462;
1&#3627408485;
1
+4&#3627408462;
0+3&#3627408462;
1+2&#3627408462;
2&#3627408485;
2
+5&#3627408462;
0+4&#3627408462;
1+3&#3627408462;
2+2&#3627408462;
3&#3627408485;
3
+⋯

&#3627408454;&#3627408470;&#3627408480;&#3627408481;&#3627408466;&#3627408474;&#3627408462; &#3627408465;&#3627408466; &#3627408440;&#3627408464;&#3627408482;&#3627408462;&#3627408464;&#3627408470;&#3627408476;&#3627408475;&#3627408466;&#3627408480;
&#3627408462;
&#3627408476;=0
3&#3627408462;
0+2&#3627408462;
1=1
4&#3627408462;
0+3&#3627408462;
1+2&#3627408462;
2=
1
4

5&#3627408462;
0+4&#3627408462;
1+3&#3627408462;
2+2&#3627408462;
3=
1
9

&#3627408462;
0=&#3627408462;→ 0
&#3627408462;
1=&#3627408463;→
1
2

&#3627408462;
2=&#3627408464;→ −
5
8

&#3627408462;
3=&#3627408465;→−
1
144


1
&#3627408475;
2
&#3627408485;
&#3627408475;∞
&#3627408475;<1
&#3627408475;+2&#3627408485;
&#3627408475;∞
&#3627408475;<0
=
1
2
&#3627408485;−
5
8
&#3627408485;
2

1
144
&#3627408485;
3
+⋯
1&#3627408485;
1
+
1
4
&#3627408485;
2
+
1
9
&#3627408485;
3
+⋯=2&#3627408462;
0&#3627408485;
0
+3&#3627408462;
0+2&#3627408462;
1&#3627408485;
1
+4&#3627408462;
0+3&#3627408462;
1+2&#3627408462;
2&#3627408485;
2
+5&#3627408462;
0+4&#3627408462;
1+3&#3627408462;
2+2&#3627408462;
3&#3627408485;
3
+⋯

Series de
Taylor

Forma de una serie de potencia convergente
&#3627408467;&#3627408485;= &#3627408462;
&#3627408475;(&#3627408485;−&#3627408464;)
&#3627408475;

&#3627408462;
&#3627408475;=
&#3627408467;
(&#3627408475;)
&#3627408464;
&#3627408475;!

&#3627408467;&#3627408485;=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;+
&#3627408467;´´(&#3627408464;)
2!
(&#3627408485;−&#3627408464;)
2
+⋯+
&#3627408467;
&#3627408475;
&#3627408464;
&#3627408475;!
&#3627408485;−&#3627408464;
&#3627408475;
+⋯

Series de Taylor y de Maclaurin

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
+⋯
Series de Taylor←??????&#3627408462;&#3627408479;&#3627408462; &#3627408485;=&#3627408464; Series de &#3627408448;&#3627408462;&#3627408464;&#3627408473;&#3627408462;&#3627408482;&#3627408479;&#3627408470;&#3627408475;←??????&#3627408462;&#3627408479;&#3627408462; &#3627408464;=0
&#3627408386;&#3627408424;&#3627408421;&#3627408418;&#3627408423; &#3627408396;&#3627408410;&#3627408412;&#3627408421;&#3627408410;&#3627408430;&#3627408427;&#3627408418;&#3627408423; &#3627408385;&#3627408427;&#3627408424;&#3627408424;&#3627408420; &#3627408403;&#3627408410;&#3627408434;&#3627408421;&#3627408424;&#3627408427;

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
+⋯

Construcción de una Serie de Potencias
Aplicar la función f(x)= Cos (x), para formar una Serie de Maclaurin

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
+⋯
&#3627409359;.

&#3627408467;
&#3627408475;
(0)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;0+&#3627408467;´0&#3627408485;

&#3627408475;<0
+
&#3627408467;
´´
(0)
2!
(&#3627408485;)
2
+
&#3627408467;
3
(0)
3!
(&#3627408485;)
3
+
&#3627408467;
4
(0)
4!
(&#3627408485;)
4


2
3
&#3627408467;&#3627408485;=cos&#3627408485; → &#3627408467;0=cos0=1
&#3627408467;´&#3627408485;=−&#3627408480;&#3627408466;&#3627408475;&#3627408485; → &#3627408467;´0=−&#3627408480;&#3627408466;&#3627408475;0=0
&#3627408467;´´&#3627408485;=−cos&#3627408485; → &#3627408467;´´0=−cos0=−1
&#3627408467;
3
&#3627408485;=&#3627408480;&#3627408466;&#3627408475;&#3627408485; → &#3627408467;
3
0=&#3627408480;&#3627408466;&#3627408475;0=0
&#3627408467;
4
&#3627408485;=cos&#3627408485; → &#3627408467;
4
0=cos0=1
&#3627408467;
5
&#3627408485;=−&#3627408480;&#3627408466;&#3627408475;&#3627408485; → &#3627408467;
5
0=−&#3627408480;&#3627408466;&#3627408475;0=0

&#3627408467;0=cos0=&#3627409359;
&#3627408467;´0=−&#3627408480;&#3627408466;&#3627408475;0=0
&#3627408467;´´0=−cos0=−1
&#3627408467;
3
0=&#3627408480;&#3627408466;&#3627408475;0=0
&#3627408467;
4
0=cos0=1
&#3627408467;
5
0=−&#3627408480;&#3627408466;&#3627408475;0=0
4

&#3627408467;
&#3627408475;
(0)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;0+&#3627408467;´0&#3627408485;

&#3627408475;<0
+
&#3627408467;
´´
(0)
2!
(&#3627408485;)
2
+
&#3627408467;
3
(0)
3!
(&#3627408485;)
3
+
&#3627408467;
4
(0)
4!
(&#3627408485;)
4


=1−
1
2!
&#3627408485;
2
+
1
4!
&#3627408485;
4

1
6!
&#3627408485;
6
+
1
8!
&#3627408485;
8
−⋯
5
&#3627408467;
&#3627408475;
(0)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=
(−1)
&#3627408475;
2&#3627408475;!
&#3627408485;
2&#3627408475;

&#3627408475;<0

&#3627408475;<0

=
(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;!

&#3627408475;<0

6
Intervalo de Convergencia

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;!

&#3627408475;<0

Intervalo de Convergencia

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;!

&#3627408475;<0

6
??????&#3627408479;&#3627408470;&#3627408481;&#3627408466;&#3627408479;&#3627408470;&#3627408476; &#3627408465;&#3627408466; &#3627408473;&#3627408462; &#3627408479;&#3627408462;&#3627408487;ó&#3627408475;
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
<1
lim
&#3627408475;→∞
(−1)
&#3627408475;:1
&#3627408485;
2(&#3627408475;:1)
2&#3627408475;+1!
(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;!
<1
lim
&#3627408475;→∞
−1
&#3627408475;:1
&#3627408485;
2&#3627408475;:1
2&#3627408475;!
2&#3627408475;+1!(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
<1
lim
&#3627408475;→∞
−1
&#3627408475;:1
&#3627408485;
2&#3627408475;:2
2&#3627408475;!
2&#3627408475;+2! (−1)
&#3627408475;
&#3627408485;
2&#3627408475;
<1
lim
&#3627408475;→∞
−1
&#3627408475;
−1
1
&#3627408485;
2&#3627408475;
&#3627408485;
2
2&#3627408475;!
−1
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;+2(2&#3627408475;+1)(2&#3627408475;)!
<1
lim
&#3627408475;→∞
−1
1
&#3627408485;
2
2&#3627408475;+2(2&#3627408475;+1)
<1
lim
&#3627408475;→∞
−&#3627408485;
2
(2&#3627408475;+2)(2&#3627408475;+1)
<1

−&#3627408485;
2
lim
&#3627408475;→∞
1
(2&#3627408475;+2)(2&#3627408475;+1)
<1
−&#3627408485;
2
(0)
lim
&#3627408475;→∞
(−1)
&#3627408475;:1
&#3627408485;
2(&#3627408475;:1)
2&#3627408475;+1!
(−1)
&#3627408475;
&#3627408485;
2&#3627408475;
2&#3627408475;!
=0
0<1
??????&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408477;&#3627408462;&#3627408479;&#3627408462; &#3627408481;&#3627408476;&#3627408465;&#3627408476; &#3627408485;→??????&#3627408475;&#3627408481;&#3627408466;&#3627408479;&#3627408483;&#3627408462;&#3627408473;&#3627408476;&#3627408479; &#3627408465;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408464;&#3627408470;&#3627408462;:(−∞,∞)

Si el lim
n→∞
R
n=0 para todo x en el intervalo I,entonces la Serie de Taylor
converge y es igual a fx.
&#3627408467;&#3627408485;=
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;

&#3627408475;<0

Convergencia de las Series de Taylor
&#3627408467;&#3627408485;=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−1+
&#3627408467;´´&#3627408464;
2!
(&#3627408485;−&#3627408464;)
2
+⋯+
&#3627408467;
&#3627408475;
&#3627408464;
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
+&#3627408453;
&#3627408475;(&#3627408475;)
&#3627408403;&#3627408414;&#3627408424;&#3627408427;&#3627408414;&#3627408422;&#3627408410; &#3627408413;&#3627408414; &#3627408403;&#3627408410;&#3627408434;&#3627408421;&#3627408424;&#3627408427;
&#3627408453;
&#3627408475;&#3627408475;=
&#3627408467;
&#3627408475;:1
&#3627408487;
(&#3627408475;+1)!
(&#3627408485;−&#3627408464;)
&#3627408475;:1

Pasos para encontrar una Serie de Taylor
Derivar f(x) varias veces y evaluar cada derivada en c
&#3627408467;&#3627408464;,&#3627408467;´&#3627408464;,&#3627408467;´´&#3627408464;,&#3627408467;
3
&#3627408464;,&#3627408467;
4
&#3627408464;+⋯+&#3627408467;
&#3627408475;
&#3627408464;+⋯
Tratar de identificar un patrón.
Usar la sucesión desarrollada en el primer paso para formar los coeficientes de
Taylor &#3627408462;
&#3627408475;=
??????
??????
(&#3627408464;)
&#3627408475;!
, y determinar el intervalo de convergencia de la serie de
potencia resultante.
&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;+
&#3627408467;´´(&#3627408464;)
2!
(&#3627408485;−&#3627408464;)
2
+⋯+
&#3627408467;
&#3627408475;
&#3627408464;
&#3627408475;!
&#3627408485;−&#3627408464;
&#3627408475;
+⋯
Dentro de este intervalo de convergencia, determinar si la serie converge o no a
f(x).
1
3
2

Excepción a las Series de Taylor y Maclaurin
&#3627408467;&#3627408485;=&#3627408485; Buscar la Serie de Taylor para c=2 y la Serie de Maclaurin a la siguiente función:
&#3627408506;&#3627408518;&#3627408531;??????&#3627408518; &#3627408517;&#3627408518; &#3627408507;??????&#3627408538;&#3627408525;&#3627408528;&#3627408531;
&#3627408467;&#3627408485;=&#3627408485; → &#3627408467;2=2=2
&#3627408467;´&#3627408485;=
&#3627408485;
&#3627408485;
→ &#3627408467;

2=
2
2
=1
&#3627408467;´´&#3627408485;=0 →&#3627408467;´´2=0
&#3627408467;
2:&#3627408475;
&#3627408485;=0→&#3627408467;
2:&#3627408475;
2=0

&#3627408467;
&#3627408475;
(2)
&#3627408475;!
(&#3627408485;−2)
&#3627408475;
=&#3627408467;2+&#3627408467;´2&#3627408485;−2

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(2)
&#3627408475;!
(&#3627408485;−2)
&#3627408475;
+⋯

&#3627408467;
&#3627408475;
(2)
&#3627408475;!
(&#3627408485;−2)
&#3627408475;
=2+&#3627408485;−2

&#3627408475;<0

??????&#3627408475;&#3627408481;&#3627408466;&#3627408479;&#3627408483;&#3627408462;&#3627408473;&#3627408476; &#3627408465;&#3627408466; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408464;&#3627408470;&#3627408462;
(−∞,∞)

&#3627408467;
&#3627408475;
(2)
&#3627408475;!
(&#3627408485;−2)
&#3627408475;
=&#3627408485;

&#3627408475;<0

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
+⋯
&#3627408506;&#3627408518;&#3627408531;??????&#3627408518; &#3627408517;&#3627408518; &#3627408500;??????&#3627408516;&#3627408525;??????&#3627408534;&#3627408531;??????&#3627408527;
&#3627408467;&#3627408485;=&#3627408485; → &#3627408467;0=0=0
&#3627408467;´&#3627408485;=
&#3627408485;
&#3627408485;
→ &#3627408467;

0=
0
0
=&#3627408449;&#3627408476; &#3627408465;&#3627408466;&#3627408467;&#3627408470;&#3627408475;&#3627408470;&#3627408465;&#3627408476;
&#3627408467;´´&#3627408485;=0 →&#3627408467;´´0=0
&#3627408467;
2:&#3627408475;
&#3627408485;=0 →&#3627408467;
2:&#3627408475;
0=0
Para la función fx=&#3627408485;,no es posible determinar la Serie de Maclaurin
Buscar la Serie de Taylor para c=2 y la Serie de Maclaurin a la siguiente función:

¿Es posible que una función cuyas derivadas existan en el punto dado,
pero que no se pueda construir una Serie de Taylor con ella?
&#3627408467;&#3627408485;=&#3627408466;
;
1
??????
2

&#3627408480;&#3627408470; &#3627408485;≠0
0 &#3627408480;&#3627408470; &#3627408485;=0

&#3627408485;≠0→&#3627408464;=2,
&#3627408467;&#3627408485;=&#3627408466;
;
1
??????
2
→&#3627408467;2=&#3627408466;
;
1
4
&#3627408467;´&#3627408485;=
2&#3627408466;
;
1
??????
2
&#3627408485;
3
→&#3627408467;
´
2=
1
4&#3627408466;
1
4

&#3627408467;
2
&#3627408485;=
2(2&#3627408466;
;
1
??????
2
−3&#3627408466;
;
1
??????
2
&#3627408485;
2
)
&#3627408485;
6
→&#3627408467;
2
2=−
5
16&#3627408466;
1
4


&#3627408397;&#3627408424; &#3627408414;&#3627408428; &#3627408425;&#3627408424;&#3627408428;&#3627408418;&#3627408411;&#3627408421;&#3627408414; &#3627408412;&#3627408424;&#3627408423;&#3627408429;&#3627408427;&#3627408418;&#3627408430;&#3627408418;&#3627408427; &#3627408430;&#3627408423;&#3627408410; &#3627408402;&#3627408414;&#3627408427;&#3627408418;&#3627408414; &#3627408413;&#3627408414;
&#3627408396;&#3627408410;&#3627408412;&#3627408421;&#3627408410;&#3627408430;&#3627408427;&#3627408418;&#3627408423; &#3627408425;&#3627408410;&#3627408427;&#3627408410; &#3627408519;&#3627408537;&#3627408516;&#3627408534;??????&#3627408527;&#3627408517;&#3627408528; &#3627408537;=&#3627409358;
=
&#3627408467;´&#3627408485;=
2&#3627408466;
;
1
??????
2
&#3627408485;
2
&#3627408480;&#3627408470; &#3627408485;≠0
0 &#3627408480;&#3627408470; &#3627408485;=0

&#3627408467;
&#3627408475;
0=0
&#3627408467;
&#3627408475;
&#3627408464;
&#3627408475;!
&#3627408485;−&#3627408464;
&#3627408475;
=

&#3627408475;<0
&#3627408466;
;
1
4+&#3627408466;
;
1
4&#3627408485;−2+
1
8&#3627408466;
1
4
&#3627408485;−2
2

5
96&#3627408466;
1
4
&#3627408485;−2
3
+⋯

Lista de Series de Potencias para Funciones Elementales
&#3627408441;&#3627408482;&#3627408475;&#3627408464;&#3627408470;ó&#3627408475; ??????&#3627408475;&#3627408481;&#3627408466;&#3627408479;&#3627408483;&#3627408462;&#3627408473;&#3627408476;
1
&#3627408485;
=1−&#3627408485;−1+(&#3627408485;−1)
2
−&#3627408485;−1
3
+&#3627408485;−1
4
−⋯+1
&#3627408475;
&#3627408485;−1
&#3627408475;
+⋯ 0<&#3627408485;<2
1
1+&#3627408485;
=1−&#3627408485;+&#3627408485;
2
−&#3627408485;
3
+&#3627408485;
4
−&#3627408485;
5
+⋯+−1
&#3627408475;
&#3627408485;
&#3627408475;
+⋯ −1<&#3627408485;<1
ln&#3627408485;=&#3627408485;−1−
&#3627408485;−1
2
2
+
&#3627408485;−1
3
3

&#3627408485;−1
4
4
+⋯+
−1
&#3627408475;;1
&#3627408485;−1
&#3627408475;
&#3627408475;
+⋯ 0<&#3627408485;≤2
&#3627408466;
??????
=1+&#3627408485;+
&#3627408485;
2
2!
+
&#3627408485;
3
3!
+
&#3627408485;
4
4!
+
&#3627408485;
5
5!
+⋯+
&#3627408485;
&#3627408475;
&#3627408475;!
+⋯ −∞<&#3627408485;<∞
&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;−
&#3627408485;
3
3!
+
&#3627408485;
5
5!

&#3627408485;
7
7!
+
&#3627408485;
9
9!
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1!
−⋯ −∞<&#3627408485;<∞
cos&#3627408485;=1−
&#3627408485;
2
2!
+
&#3627408485;
4
4!

&#3627408485;
6
6!
+
&#3627408485;
8
8!
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;!
+⋯ −∞<&#3627408485;<∞
arctan&#3627408485;=&#3627408485;−
&#3627408485;
3
3
+
&#3627408485;
5
5

&#3627408485;
7
7
+
&#3627408485;
9
9
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1
−⋯. −1≤&#3627408485;≤1
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;+
&#3627408485;
3
2∙3
+
1∙3&#3627408485;
5
2∙4∙5
+
1∙3∙5&#3627408485;
7
2∙4∙6∙7
+⋯+
2&#3627408475;!&#3627408485;
2&#3627408475;:1
(2
&#3627408475;
&#3627408475;!)
&#3627408475;
(2&#3627408475;+1)
+⋯ −1≤&#3627408485;≤1

(1+&#3627408485;)
&#3627408472;
=1+kx+
&#3627408472;&#3627408472;−1&#3627408485;
2
2!
+
&#3627408472;(&#3627408472;−1)(&#3627408472;−2)&#3627408485;
3
3!
+
&#3627408472;&#3627408472;−1&#3627408472;−2&#3627408472;−3&#3627408485;
4
4!
−1≤&#3627408485;≤1

En los ejercicios 53 y 57, encontrar el intervalo de
convergencia de la serie de potencias. (Asegurarse de
incluir una verificación para la convergencia en los
puntos terminales del intervalo)
Repaso de la unidad 7, pág.503. Essential Calculus, Larson.

Aplicar un Criterio para determinar la convergencia de la serie. (Raíz, cociente)
Buscar el límite resultante
Determinar el intervalo y el radio de convergencia
Evaluar la serie en los extremos del intervalo
Determinar el verdadero intervalo(tomado en cuenta la convergencia de los extremos)
Comprobar los resultados
Pasos para realizar los ejercicios
1
2
3
6
4
5

&#3627409363;&#3627409361;)
&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;

&#3627408527;<&#3627409358;

&#3627409363;&#3627409361;)
&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;

&#3627408527;<&#3627409358;

Criterio de la raíz
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
<1 lim
&#3627408475;→∞
&#3627408485;
10
&#3627408475;??????
<1

lim
&#3627408475;→∞
&#3627408485;
10
&#3627408475;
1
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;
&#3627408475;
10
&#3627408475;
1
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;
&#3627408475;
1
&#3627408475;
10
&#3627408475;
1
&#3627408475;
<1
&#3627408462;
&#3627408474;
??????
=&#3627408462;
&#3627408474;
&#3627408475;
&#3627408462;
&#3627408463;
&#3627408474;
=
&#3627408462;
&#3627408474;
&#3627408463;
&#3627408474;

(&#3627408485;
&#3627408462;
)
&#3627408463;
=&#3627408485;
&#3627408462;(&#3627408463;)

&#3627408401;&#3627408410;&#3627408435;ó&#3627408423; &#3627408387;&#3627408414;&#3627408428;&#3627408410;&#3627408427;&#3627408427;&#3627408424;&#3627408421;&#3627408421;&#3627408424;

lim
&#3627408475;→∞
&#3627408485;
1
10
1
<1
lim
&#3627408475;→∞
&#3627408485;
10
&#3627408475;??????
<1→lim
&#3627408475;→∞
&#3627408485;
&#3627408475;
1
&#3627408475;
10
&#3627408475;
1
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;
10
<1
&#3627408485;
10
<1
lim
&#3627408475;→∞
&#3627408485;
10
&#3627408475;??????
=
&#3627408485;
10

&#3627408475;
1
&#3627408475;
=
&#3627408475;
&#3627408475;
=1
lim
??????→&#3627408464;
??????=??????

&#3627408525;??????&#3627408526;
&#3627408527;→∞
&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;&#3627408527;
=
&#3627408537;
&#3627409359;&#3627409358;

Criterio de la raíz
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;
??????
<1
??????&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408480;&#3627408470;
&#3627408485;
10
<1
&#3627408485;
10
(10)<1(10)
&#3627408485;<10
&#3627408453;=10
−10<&#3627408485;<10
Intervalo de Convergencia: (−10,10)

&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;

&#3627408527;<&#3627409358;
(−10,10)

&#3627408485;
10
&#3627408475;
=

&#3627408475;<0

−10
10
&#3627408475;∞
&#3627408475;<0

= −1
&#3627408475;

&#3627408475;<0

−1
&#3627408475;

&#3627408475;<0
=(−1)
0
+−1
1
+(−1)
2
+(−1)
3
+−1
4
+⋯
−1
&#3627408475;

&#3627408475;<0
=1−1+1−1+1−⋯

&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;

&#3627408527;<&#3627409358;
Diverge para x=−10
lim
&#3627408475;→∞
(−1)
&#3627408475;
=&#3627408449;&#3627408476; &#3627408466;&#3627408485;&#3627408470;&#3627408480;&#3627408481;&#3627408466;
&#3627408462;
&#3627408462;
=1
&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
=&#3627408462;
0+&#3627408462;
1&#3627408485;+&#3627408462;
2&#3627408485;
2
+&#3627408462;
3&#3627408485;
3
+⋯+&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
+⋯

&#3627408475;<0

Criterio del Término n−símo para la divergencia
&#3627408480;&#3627408470; lim
&#3627408475;→∞
&#3627408462;
&#3627408475;≠0→ &#3627408462;
&#3627408475; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;

&#3627408475;<1

&#3627408485;
10
&#3627408475;
=

&#3627408475;<0

10
10
&#3627408475;∞
&#3627408475;<0

= 1
&#3627408475;

&#3627408475;<0

1
&#3627408475;

&#3627408475;<0
=(1)
0
+1
1
+(1)
2
+(1)
3
+1
4
+⋯
1
&#3627408475;

&#3627408475;<0
=1+1+1+1+1+⋯
Diverge para x=10
Intervalo de Convergencia de
&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;
&#3627408518;&#3627408532;:

&#3627408527;<&#3627409358;
(−10,10)

&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;

&#3627408527;<&#3627409358;

&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
=&#3627408462;
0+&#3627408462;
1&#3627408485;+&#3627408462;
2&#3627408485;
2
+&#3627408462;
3&#3627408485;
3
+⋯+&#3627408462;
&#3627408475;&#3627408485;
&#3627408475;
+⋯

&#3627408475;<0

&#3627408462;
&#3627408462;
=1

Comprobación
Intervalo de Convergencia de
&#3627408537;
&#3627409359;&#3627409358;
&#3627408527;
&#3627408518;&#3627408532;:

&#3627408527;<&#3627409358;
(−10,10)

&#3627409363;&#3627409365;) &#3627408527;!(&#3627408537;−&#3627409360;)
&#3627408527;

&#3627408527;<&#3627409358;

&#3627408527;!(&#3627408537;−&#3627409360;)
&#3627408527;

&#3627408527;<&#3627409358;

??????&#3627408479;&#3627408470;&#3627408481;&#3627408466;&#3627408479;&#3627408470;&#3627408476; &#3627408465;&#3627408466;&#3627408473; ??????&#3627408476;&#3627408464;&#3627408470;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408475;+1!(&#3627408485;−2)
&#3627408475;:1
&#3627408475;!(&#3627408485;−2)
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408475;+1&#3627408475;!(&#3627408485;−2)
&#3627408475;
(&#3627408485;−2)
1
&#3627408475;!(&#3627408485;−2)
&#3627408475;
<1
lim
&#3627408475;→∞
(&#3627408475;+1)(&#3627408485;−2)<1
lim
&#3627408475;→∞
(&#3627408475;+1)&#3627408485;−2<1
&#3627408485;−2lim
&#3627408475;→∞
(&#3627408475;+1)<1
&#3627408462;
&#3627408474;
&#3627408462;
&#3627408475;
=&#3627408462;
&#3627408474;:&#3627408475;

&#3627408462;
&#3627408462;
=1
lim
??????→&#3627408464;
??????&#3627408485;=??????lim
??????→&#3627408464;
&#3627408485;

&#3627408485;−2lim
&#3627408475;→∞
(&#3627408475;+1)<1 lim
&#3627408475;→∞
&#3627408475;+1!(&#3627408485;−2)
&#3627408475;:1
&#3627408475;!(&#3627408485;−2)
&#3627408475;

&#3627408485;−2lim
&#3627408475;→∞
&#3627408475;+1=∞ Condición de Convergencia de una
Serie de Potencias:
1.&#3627408454;&#3627408476;&#3627408473;&#3627408476; &#3627408464;&#3627408476;&#3627408475;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466; &#3627408466;&#3627408475; &#3627408482;&#3627408475; ??????&#3627408482;&#3627408475;&#3627408481;&#3627408476;,&#3627408453;=0
lim
&#3627408475;→∞
&#3627408475;+1!(&#3627408485;−2)
&#3627408475;:1
&#3627408475;!(&#3627408485;−2)
&#3627408475;
=∞
&#3627408475;!(&#3627408485;−2)
&#3627408475;
solo converge en x=2

&#3627408475;<0

Comprobación
&#3627408475;!(&#3627408485;−2)
&#3627408475;
solo converge en x=2

&#3627408475;<0

En el ejercicio 61, encontrar una serie
geométrica de potencia centrada en 0 para la
función
Repaso de la unidad 7, pág.503. Essential Calculus, Larson.

Comprobar si la función dada corresponde a la forma de la serie geométrica

Adaptar la función (si es necesario) a la forma
??????
&#3627409359;;&#3627408531;

Determinar a y r
Sustituir las variables de la serie geométrica por los valores obtenidos
Comprobar los resultados
Pasos para realizar el ejercicio
1
2
3
5
4

&#3627409364;&#3627409359;.&#3627408520;&#3627408537;=
&#3627409360;
&#3627409361;−&#3627408537;

&#3627408520;&#3627408537;=
&#3627409360;
&#3627409361;−&#3627408537;


2
3−&#3627408485;
=
2
3
3
3

&#3627408485;
3

&#3627408467;&#3627408485;=
&#3627408462;
1−&#3627408485;

=
2
3
1−
&#3627408485;
3

&#3627408467;&#3627408485;=
&#3627408462;
1−&#3627408485;
&#3627408462;=
2
3
,&#3627408479;=
&#3627408485;
3

2
3−&#3627408485;
=
2
3
&#3627408485;
3

&#3627408475;

&#3627408475;<0

&#3627408506;&#3627408518;&#3627408531;??????&#3627408518; &#3627408494;&#3627408518;&#3627408528;&#3627408526;é&#3627408533;&#3627408531;??????&#3627408516;??????
??????&#3627408531;
&#3627408527;
=
??????
&#3627409359;−&#3627408531;

&#3627408527;<&#3627409358;

&#3627408462;
&#3627408462;
=1

2
3
&#3627408485;
3

&#3627408475;

&#3627408475;<0

??????&#3627408479;&#3627408470;&#3627408481;&#3627408466;&#3627408479;&#3627408470;&#3627408476; &#3627408465;&#3627408466;&#3627408473; ??????&#3627408476;&#3627408464;&#3627408470;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
lim
&#3627408475;→∞
&#3627408462;
&#3627408475;:1
&#3627408462;
&#3627408475;
<1

2
3
&#3627408485;
3

&#3627408475;
=
2
3
&#3627408485;
&#3627408475;
3
&#3627408475;

&#3627408475;<0

&#3627408475;<0

=
2&#3627408485;
&#3627408475;
3
1:&#3627408475;

&#3627408475;<0

lim
&#3627408475;→∞
2&#3627408485;
&#3627408475;:1
3
1:&#3627408475;:1
2&#3627408485;
&#3627408475;
3
1:&#3627408475;
<1
lim
&#3627408475;→∞
2&#3627408485;
&#3627408475;:1
3
1:&#3627408475;
3
2:&#3627408475;
2&#3627408485;
&#3627408475;
<1
lim
&#3627408475;→∞
&#3627408485;
3
<1
&#3627408485;
3
<1
|&#3627408485;|<3
&#3627408462;
&#3627408463;
&#3627408464;
=
&#3627408462;
&#3627408464;
&#3627408463;
&#3627408464;

&#3627408485;
&#3627408462;
(&#3627408485;)
&#3627408463;
=&#3627408485;
&#3627408462;:&#3627408463;

&#3627408462;
&#3627408463;
&#3627408464;
&#3627408465;
=
&#3627408462;&#3627408465;
&#3627408463;&#3627408464;

2&#3627408485;
&#3627408475;:1
2&#3627408485;
&#3627408475;
=&#3627408485;,
3
1:&#3627408475;
3
2:&#3627408475;
=
1
3

lim
&#3627408475;→∞
??????=??????
←&#3627408505;=&#3627409361;
−3<&#3627408485;<3
Intervalo de convergencia(−3,3)
&#3627408485;=−3

2
3
−1
&#3627408475;

&#3627408475;<0

??????&#3627408528;&#3627408527;&#3627408535;&#3627408518;&#3627408531;&#3627408520;&#3627408518;&#3627408527;&#3627408516;???????????? &#3627408517;&#3627408518; &#3627408534;&#3627408527;?????? &#3627408506;&#3627408518;&#3627408531;??????&#3627408518; &#3627408494;&#3627408518;&#3627408528;&#3627408526;é&#3627408533;&#3627408531;??????&#3627408516;??????
&#3627409358;<|&#3627408531;|<&#3627409359;
−1=1

2
3
−1
&#3627408475;

&#3627408475;<0
&#3627408466;&#3627408480; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
&#3627408485;=3

2
3
1
&#3627408475;

&#3627408475;<0

1=1

2
3
1
&#3627408475;

&#3627408475;<0
&#3627408466;&#3627408480; &#3627408465;&#3627408470;&#3627408483;&#3627408466;&#3627408479;&#3627408468;&#3627408466;&#3627408475;&#3627408481;&#3627408466;
&#3627408479;=−1 &#3627408479;=1
&#3627408392;&#3627408423;&#3627408429;&#3627408414;&#3627408427;&#3627408431;&#3627408410;&#3627408421;&#3627408424; &#3627408413;&#3627408414; &#3627408412;&#3627408424;&#3627408423;&#3627408431;&#3627408414;&#3627408427;??????&#3627408414;&#3627408423;&#3627408412;&#3627408418;&#3627408410;(−&#3627409361;,&#3627409361;)

Comprobación
&#3627409364;&#3627409359;)&#3627408520;&#3627408537;=
&#3627409360;
&#3627409361;−&#3627408537;
=
&#3627409360;
&#3627409361;
&#3627408537;
&#3627409361;

&#3627408527;

&#3627408527;<&#3627409358;

En el ejercicio 71, encontrar la serie de
potencia para la función centrada en c

Repaso de la unidad 7, pág.503. Essential Calculus, Larson.

Buscar la forma general de la serie de potencia que se utilizará
Derivar la función dada varias veces(el número de derivadas, es a
libre elección)
Sustituir en la fórmula general con los valores obtenidos
Comprobar los resultados
Pasos para realizar el ejercicio
1
2
3
4

&#3627409365;&#3627409359;.&#3627408494;&#3627408537;=&#3627409359;+&#3627408537;
&#3627409363;
,&#3627408516;=&#3627409358;

&#3627408467;
&#3627408475;
(0)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;0+&#3627408467;´0&#3627408485;

&#3627408475;<0
+
&#3627408467;
´´
(0)
2!
(&#3627408485;)
2
+
&#3627408467;
3
(0)
3!
(&#3627408485;)
3
+
&#3627408467;
4
(0)
4!
(&#3627408485;)
4

&#3627408494;&#3627408537;=&#3627409359;+&#3627408537;
&#3627409363;
,&#3627408516;=&#3627409358;
&#3627408442;&#3627408485;=1+&#3627408485;
5

1+&#3627408485;
5
=(1+&#3627408485;)
1
5
&#3627408467;&#3627408485; =(1+&#3627408485;)
1
5 → &#3627408467;0=(1+0)
1
5=1
&#3627408467;

&#3627408485; =
1
5
(1+&#3627408485;)
;
4
5 → &#3627408467;´0=
1
5
(1+0)
;
4
5=
1
5

&#3627408467;´´&#3627408485; =−
4
25
1+&#3627408485;
;
9
5 → &#3627408467;´´0=−
4
25
1+0
;
9
5=−
4
25

&#3627408467;
3
&#3627408485;=
36
125
1+&#3627408485;
;
14
5 → &#3627408467;
3
(0)=
36
125
(1+0)
;
14
5=
36
125

&#3627408467;
4
&#3627408485;=−
504
625
1+&#3627408485;
;
19
5 → &#3627408467;
4
0=−
504
625
1+0
;
19
5=−
504
625

&#3627408462;
&#3627408474;
??????
=&#3627408462;
&#3627408475;
&#3627408474;
1
&#3627408475;
=1
&#3627408465;
&#3627408465;&#3627408485;
&#3627408467;&#3627408468;&#3627408485;=&#3627408467;

&#3627408468;&#3627408485;&#3627408468;´&#3627408485;,
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;
&#3627408475;
=&#3627408475;&#3627408485;
&#3627408475;;1
,
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;=1,
1
5

4
5
=−
4
25
. −
4
5
−1=−
9
5


4
25

9
5
=
36
125
,−
9
5
−1=−
14
5

1
5
−1=−
4
5

36
125

14
5
=−
504
625
,−
14
5
−1=
19
5

&#3627408467;0=(1+0)
1
5 = 1
&#3627408467;´0=
1
5
(1+0)
;
4
5 =
1
5

&#3627408467;´´0=−
4
25
1+0
;
9
5 = −
4
25

&#3627408467;
3
0=
36
125
1+0
;
14
5 =
36
125

&#3627408467;
4
0=−
504
625
1+0
;
19
5 =−
504
625


&#3627408467;
&#3627408475;
(0)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;0+&#3627408467;´0&#3627408485;

&#3627408475;<0
+
&#3627408467;
´´
(0)
2!
(&#3627408485;)
2
+
&#3627408467;
3
(0)
3!
(&#3627408485;)
3
+
&#3627408467;
4
(0)
4!
(&#3627408485;)
4

=1+
1
5
&#3627408485;+−
4
252!
&#3627408485;
2
+
36
125(3!)
&#3627408485;
3
+−
504
6254!
&#3627408485;
4
+⋯
=1+
1
5
&#3627408485;−
2
25
&#3627408485;
2
+
6
125
&#3627408485;
3

21
625
&#3627408485;
4
+⋯
1+&#3627408485;
5
=1+
1
5
&#3627408485;−
2
25
&#3627408485;
2
+
6
125
&#3627408485;
3

21
625
&#3627408485;
4
+⋯
2!=2 ,
4
252
=
4
50
=
2
25

3!=6,
36
125(6)
=
36
750
=
6
125

4!=24,−
504
62524
=−
504
15,000
=−
21
625

&#3627408467;&#3627408485;=(1+&#3627408485;)
&#3627408472;

??????
&#3627408472;
=
??????(??????−1)(??????−2)…??????−k+1
&#3627408472;!

??????
0
=1 =1+
1
5
&#3627408485;+
1
5
1
5
−1
2!
&#3627408485;
2
+
1
5
1
5
−1
1
5
−2
3!
&#3627408485;
3
+⋯
??????
1
=
??????
1!

??????
3
=
??????(??????−1)(??????−2)
3!

??????
2
=
??????(??????−1)
2!

=
1
5
0
&#3627408485;
0
+
1
5
1
&#3627408485;
1
+
1
5
2
&#3627408485;
2
+
1
5
3
&#3627408485;
3
+⋯

&#3627409359;
&#3627409363;
&#3627408527;
&#3627408537;
&#3627408527;

&#3627408527;<&#3627409358;

=1+
1
5
&#3627408485;−
2
25
&#3627408485;
2
+
6
125
&#3627408485;
3
+⋯

Comprobación
&#3627408494;&#3627408537;=&#3627409359;+&#3627408537;
&#3627409363;
,&#3627408516;=&#3627409358;→1+
1
5
&#3627408485;−
2
25
&#3627408485;
2
+
6
125
&#3627408485;
3

21
625
&#3627408485;
4
+⋯

En los ejercicios 75 y 77, encontrar la suma de las
series convergentes utilizando una función muy
conocida. Identificar la función y explicar cómo se
obtuvo la suma.
Repaso de la unidad 7, pág.503. Essential Calculus, Larson.

&#3627409365;&#3627409363;.
&#3627409359;
&#3627409360;
&#3627408527;
&#3627408527;!

&#3627408527;<&#3627409358;

&#3627409359;
&#3627409360;
&#3627408527;
&#3627408527;!

&#3627408527;<&#3627409358;


1
2
&#3627408475;
&#3627408475;!
=
1
2
&#3627408475;
1
&#3627408475;!

&#3627408475;<0


&#3627408475;<0

=
1
2
&#3627408475;
1
&#3627408475;!

&#3627408475;<0

&#3627408462;
&#3627408485;&#3627408486;
=
&#3627408462;
&#3627408485;
&#3627408462;
&#3627408486;

&#3627408485;
&#3627408462;
&#3627408486;
&#3627408462;
=
&#3627408485;
&#3627408486;
&#3627408462;

Lista de Series de Potencias para Funciones Elementales
&#3627408441;&#3627408482;&#3627408475;&#3627408464;&#3627408470;ó&#3627408475; ??????&#3627408475;&#3627408481;&#3627408466;&#3627408479;&#3627408483;&#3627408462;&#3627408473;&#3627408476;
1
&#3627408485;
=1−&#3627408485;−1+(&#3627408485;−1)
2
−&#3627408485;−1
3
+&#3627408485;−1
4
−⋯+.1
&#3627408475;
&#3627408485;−1
&#3627408475;
+⋯ 0<&#3627408485;<2
1
1+&#3627408485;
=1−&#3627408485;+&#3627408485;
2
−&#3627408485;
3
+&#3627408485;
4
−&#3627408485;
5
+⋯+−1
&#3627408475;
&#3627408485;
&#3627408475;
+⋯ −1<&#3627408485;<1
ln&#3627408485;=&#3627408485;−1−
&#3627408485;−1
2
2
+
&#3627408485;−1
3
3

&#3627408485;−1
4
4
+⋯+
−1
&#3627408475;;1
&#3627408485;−1
&#3627408475;
&#3627408475;
+⋯ 0<&#3627408485;≤2
&#3627408466;
??????
=1+&#3627408485;+
&#3627408485;
2
2!
+
&#3627408485;
3
3!
+
&#3627408485;
4
4!
+
&#3627408485;
5
5!
+⋯+
&#3627408485;
&#3627408475;
&#3627408475;!
+⋯ −∞<&#3627408485;<∞
&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;−
&#3627408485;
3
3!
+
&#3627408485;
5
5!

&#3627408485;
7
7!
+
&#3627408485;
9
9!
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1!
−⋯ −∞<&#3627408485;<∞
cos&#3627408485;=1−
&#3627408485;
2
2!
+
&#3627408485;
4
4!

&#3627408485;
6
6!
+
&#3627408485;
8
8!
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;!
+⋯ −∞<&#3627408485;<∞
arctan&#3627408485;=&#3627408485;−
&#3627408485;
3
3
+
&#3627408485;
5
5

&#3627408485;
7
7
+
&#3627408485;
9
9
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1
−⋯ −1≤&#3627408485;≤1
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;+
&#3627408485;
3
2∙3
+
1∙3&#3627408485;
5
2∙4∙5
+
1∙3∙5&#3627408485;
7
2∙4∙6∙7
+⋯+
2&#3627408475;!&#3627408485;
2&#3627408475;:1
(2
&#3627408475;
&#3627408475;!)
&#3627408475;
(2&#3627408475;+1)
+⋯ −1≤&#3627408485;≤1

(1+&#3627408485;)
&#3627408472;
=1+kx+
&#3627408472;&#3627408472;−1&#3627408485;
2
2!
+
&#3627408472;(&#3627408472;−1)(&#3627408472;−2)&#3627408485;
3
3!
+
&#3627408472;&#3627408472;−1&#3627408472;−2&#3627408472;−3&#3627408485;
4
4!
−1≤&#3627408485;≤1

&#3627409359;
&#3627409360;
&#3627408527;
&#3627408527;!

&#3627408527;<&#3627409358;


1
2
&#3627408475;
&#3627408475;!
=
1
2
&#3627408475;
1
&#3627408475;!

&#3627408475;<0


&#3627408475;<0

=
1
2
&#3627408475;
1
&#3627408475;!

&#3627408475;<0

&#3627408466;
??????
=1+&#3627408485;+
&#3627408485;
2
2!
+
&#3627408485;
3
3!
+
&#3627408485;
4
4!
+
&#3627408485;
5
5!
+⋯+
&#3627408485;
&#3627408475;
&#3627408475;!
+⋯
&#3627408466;
??????
=
&#3627408485;
&#3627408475;
&#3627408475;!

&#3627408475;<0

&#3627408462;
&#3627408485;&#3627408486;
=
&#3627408462;
&#3627408485;
1
&#3627408486;

&#3627408485;
&#3627408475;
=
1
2
&#3627408475;


1
2
&#3627408475;
&#3627408475;!
=&#3627408466;
1
2

&#3627408475;<0

&#3627408466;
1
2≈1.65
=
1
2
&#3627408475;
1
&#3627408475;!

&#3627408475;<0

Comprobación
75.
1
2
&#3627408475;
&#3627408475;!
=&#3627408466;
1
2

&#3627408475;<0
&#3627408466;
1
2=&#3627408466;

&#3627409365;&#3627409365;. (−&#3627409359;)
&#3627408527;
&#3627409360;
&#3627409360;&#3627408527;
&#3627409361;
&#3627409360;&#3627408527;
&#3627409360;&#3627408527;!

&#3627408527;<&#3627409358;

cos&#3627408485;=1−
&#3627408485;
2
2!
+
&#3627408485;
4
4!

&#3627408485;
6
6!
+
&#3627408485;
8
8!
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;!
+⋯
(−&#3627409359;)
&#3627408527;
&#3627409360;
&#3627409360;&#3627408527;
&#3627409361;
&#3627409360;&#3627408527;
&#3627409360;&#3627408527;!

&#3627408527;<&#3627409358;

(−1)
&#3627408475;
2
2&#3627408475;
3
2&#3627408475;
2&#3627408475;!
= −1
&#3627408475;
2
2&#3627408475;
3
2&#3627408475;
1
2&#3627408475;!

&#3627408475;<0

&#3627408475;<0

= −1
&#3627408475;
2
3
2&#3627408475;
1
2&#3627408475;!

&#3627408475;<0
cos&#3627408485;=
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;!

&#3627408475;<0

(−1)
&#3627408475;
2
2&#3627408475;
3
2&#3627408475;
2&#3627408475;!
=cos (
2
3
)

&#3627408475;<0

&#3627408485;
&#3627408462;&#3627408463;
=
&#3627408485;
&#3627408462;
1
&#3627408463;
,
&#3627408462;
&#3627408463;
&#3627408474;
=
&#3627408462;
&#3627408474;
&#3627408463;
&#3627408474;

cos
2
3
≈0.786

Comprobación
77. (−1)
&#3627408475;
2
2&#3627408475;
3
2&#3627408475;
2&#3627408475;!
=cos (
2
3
)

&#3627408475;<0

cos
2
3
≈0.786

En los ejercicios 85 y 86, usar una serie de potencias
para encontrar el límite (si existe). Verificar el
resultado usando la regla de L´Hopital.

Repaso de la unidad 7, pág.503. Essential Calculus, Larson.

Encontrar las series de las funciones implicadas
Simplificar (de ser posible)
Resolver el límite
Comprobar con L´Hopital
Comprobar los resultados
Pasos para realizar los ejercicios
1
2
3
5
4

&#3627409366;&#3627409363;.&#3627408421;&#3627408418;&#3627408422;
&#3627408537;→&#3627409358;
+
&#3627408410;&#3627408427;&#3627408412;&#3627408429;&#3627408410;&#3627408423; (&#3627408537;)
&#3627408537;

&#3627408421;&#3627408418;&#3627408422;
&#3627408537;→&#3627409358;
+
??????&#3627408531;&#3627408516;&#3627408533;??????&#3627408527; (&#3627408537;)
&#3627408537;

&#3627408485;=&#3627408485;
1
2
lim
??????→0
+
arctan (&#3627408485;)
&#3627408485;
=lim
??????→0
+

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1

&#3627408475;<0
&#3627408485;
1
2

=lim
??????→0
+

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;:1
(2&#3627408475;+1)(&#3627408485;
1
2)

&#3627408475;<0

=lim
??????→0
+

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;:
1
2
2&#3627408475;+1

&#3627408475;<0

&#3627408462;
&#3627408463;
&#3627408464;
=
&#3627408462;
&#3627408463;&#3627408464;

&#3627408485;
2&#3627408475;:1
&#3627408485;
1
2
= &#3627408485;
4&#3627408475;:1
2
arctan&#3627408485;=&#3627408485;−
&#3627408485;
3
3
+
&#3627408485;
5
5

&#3627408485;
7
7
+
&#3627408485;
9
9
−⋯+
−1
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1
−⋯
=
(−1)
&#3627408475;
(0)
2&#3627408475;:
1
2
2&#3627408475;+1

&#3627408475;<0

=&#3627408485;
2&#3627408475;:
1
2
−&#3627409359;≤&#3627408537;≤&#3627409359;

lim
??????→0
+
&#3627408462;&#3627408479;&#3627408464;&#3627408481;&#3627408462;&#3627408475;(&#3627408485;)
&#3627408485;
=0
= 0

&#3627408475;<0

&#3627408386;&#3627408424;&#3627408422;&#3627408425;&#3627408427;&#3627408424;&#3627408411;&#3627408410;&#3627408412;&#3627408418;ó&#3627408423; &#3627408412;&#3627408424;&#3627408423; &#3627408421;&#3627408410; &#3627408401;&#3627408414;??????&#3627408421;&#3627408410; &#3627408413;&#3627408414; &#3627408395;´&#3627408391;&#3627408424;&#3627408425;&#3627408418;&#3627408429;&#3627408410;&#3627408421;
lim
??????→0
+
&#3627408462;&#3627408479;&#3627408464;&#3627408481;&#3627408462;&#3627408475;(&#3627408485;)
&#3627408485;
=lim
??????→0
+
&#3627408465;
&#3627408465;&#3627408485;
(arctan&#3627408485;)
&#3627408465;
&#3627408465;&#3627408485;
(&#3627408485;)

&#3627408505;&#3627408518;&#3627408520;&#3627408525;?????? &#3627408517;&#3627408518; &#3627408499;´&#3627408495;&#3627408528;&#3627408529;??????&#3627408533;??????&#3627408525;
lim
??????→&#3627408464;
&#3627408467;(&#3627408485;)
&#3627408468;(&#3627408485;)
=lim
??????→&#3627408464;
&#3627408467;´(&#3627408485;)
&#3627408468;´(&#3627408485;)

=lim
??????→0
+
1
&#3627408485;
2
+1
1
2&#3627408485;

&#3627408465;
&#3627408465;&#3627408482;
arctan&#3627408482;=
1
&#3627408482;
2
+1

&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;=
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;
1
2=
1
2
&#3627408485;
;
1
2
lim
??????→0
+
2&#3627408485;
&#3627408485;
2
+1

=
20
0
2
+1

lim
??????→0
+
2&#3627408485;
&#3627408485;
2
+1
=0
&#3627408462;
&#3627408464;
&#3627408463;
&#3627408465;
=
&#3627408462;&#3627408465;
&#3627408463;&#3627408464;

lim
??????→0
+
&#3627408462;&#3627408479;&#3627408464;&#3627408481;&#3627408462;&#3627408475;(&#3627408485;)
&#3627408485;
=0
lim
??????→0
+

(−1)
&#3627408475;
&#3627408485;
2&#3627408475;:1
2&#3627408475;+1

&#3627408475;<0
&#3627408485;
1
2
=
(−1)
&#3627408475;
(0)
2&#3627408475;:
1
2
2&#3627408475;+1

&#3627408475;<0

Comprobación
&#3627409366;&#3627409363;.&#3627408421;&#3627408418;&#3627408422;
&#3627408537;→&#3627409358;
+
??????&#3627408531;&#3627408516;&#3627408533;??????&#3627408527;(&#3627408537;)
&#3627408537;
=&#3627409358;

&#3627409366;&#3627409364;.lim
&#3627408537;→&#3627409358;
??????&#3627408531;&#3627408516;&#3627408532;&#3627408518;&#3627408527;(&#3627408537;)
&#3627408537;

&#3627409366;&#3627409364;.lim
&#3627408537;→&#3627409358;
??????&#3627408531;&#3627408516;&#3627408532;&#3627408518;&#3627408527;(&#3627408537;)
&#3627408537;

&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;+
&#3627408485;
3
2∙3
+
1∙3&#3627408485;
5
2∙4∙5
+
1∙3∙5&#3627408485;
7
2∙4∙6∙7
+⋯
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475; &#3627408485;=&#3627408485;+
&#3627408485;
3
6
+
3&#3627408485;
5
40
+
15&#3627408485;
7
336
+⋯
&#3627408485;
lim
??????→0
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;(&#3627408485;)
&#3627408485;
=lim
??????→0
&#3627408485;+
&#3627408485;
3
6
+
3&#3627408485;
5
40
+
15&#3627408485;
7
336
+⋯
&#3627408485;

=1+
0
2
6
+
3(0)
4
40
+
5(0)
6
112
+⋯
&#3627408462;
&#3627408462;
=1,
=lim
??????→0
&#3627408485;
&#3627408485;
+
&#3627408485;
3
6
&#3627408485;
+
3&#3627408485;
5
40
&#3627408485;
+
15&#3627408485;
7
336
&#3627408485;
+⋯
&#3627408462;
&#3627408464;
&#3627408463;
&#3627408465;
=
&#3627408462;&#3627408465;
&#3627408463;&#3627408464;

&#3627408462;
&#3627408485;
+
&#3627408463;
&#3627408485;
=
&#3627408462;+&#3627408463;
&#3627408485;

=lim
??????→0
1+
&#3627408485;
2
6
+
3&#3627408485;
4
40
+
5&#3627408485;
6
112
+⋯
=1
lim
??????→0
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;(&#3627408485;)
&#3627408485;
=1
&#3627408386;&#3627408424;&#3627408422;&#3627408425;&#3627408427;&#3627408424;&#3627408411;&#3627408410;&#3627408412;&#3627408418;ó&#3627408423; &#3627408412;&#3627408424;&#3627408423; &#3627408401;&#3627408414;??????&#3627408421;&#3627408410; &#3627408413;&#3627408414; &#3627408395;´&#3627408391;&#3627408424;&#3627408425;&#3627408418;&#3627408429;&#3627408410;&#3627408421;
lim
??????→0
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;(&#3627408485;)
&#3627408485;
=lim
??????→0
&#3627408465;
&#3627408465;&#3627408485;
(&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;&#3627408485;)
&#3627408465;
&#3627408465;&#3627408485;
(&#3627408485;)

=lim
??????→0
1
1−&#3627408485;
2
1

=lim
??????→0
1
1−&#3627408485;
2

=1
=
1
1−0
2

&#3627408462;
&#3627408464;
&#3627408463;
&#3627408465;
=
&#3627408462;&#3627408465;
&#3627408463;&#3627408464;

lim
??????→&#3627408464;
&#3627408467;(&#3627408485;)
&#3627408468;(&#3627408485;)
=lim
??????→&#3627408464;
&#3627408467;´(&#3627408485;)
&#3627408468;´(&#3627408485;)

&#3627408465;
&#3627408465;&#3627408485;
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;&#3627408485;=
1
1−&#3627408485;
2
,
&#3627408465;
&#3627408465;&#3627408485;
&#3627408485;=1
1=1,
1
1
=1
−&#3627409359;≤&#3627408537;≤&#3627409359;

Comprobación
lim
??????→0
&#3627408462;&#3627408479;&#3627408464;&#3627408480;&#3627408466;&#3627408475;(&#3627408485;)
&#3627408485;
=1

Ejercicios
Encuentra el Intervalo de
Convergencia

(−1)
&#3627408475;
(&#3627408485;−2)
&#3627408475;
(&#3627408475;+1)
2

&#3627408475;<0

Encontrar una serie geométrica de
potencia centrada en 0 para la
función
&#3627408469;&#3627408485;=
7
4+&#3627408485;

Encontrar la serie de potencia para la
función centrada en c
&#3627408467;&#3627408485;=
1
&#3627408485;
,&#3627408464;=−1
Encuentra la función que define la serie
utilizando una serie de una función
conocida
(−1)
&#3627408475;:1
1
4
&#3627408475;
&#3627408475;

&#3627408475;<1

Conclusión

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;−&#3627408464;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;−&#3627408464;)
&#3627408475;
+⋯

&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
=&#3627408467;&#3627408464;+&#3627408467;´&#3627408464;&#3627408485;

&#3627408475;<0
+⋯+
&#3627408467;
&#3627408475;
(&#3627408464;)
&#3627408475;!
(&#3627408485;)
&#3627408475;
+⋯

Frase Motivacional
«La mente es como un jardín, lo
que siembras en ellas, lo cosechas
en la vida».
Proverbio chino

Recursos

Bibliografía
Larson, R; Edwards, B.H. (2010). Cálculo de una Variable,
Novena Edición. McGRAW-HILL. Consultado: 04 al 14 de
Nov. 2023.

Larson, R; Edwards, B.H; Hostetler, R. (2008). Essential
Calculus: Early Trascendental Function. Houghton Mifflin
Company. Consultado; 04 al 14 de Nov. 2023.

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