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About This Presentation
Engineering mathematics notes
Slide Content
Engineering Mathematics-I
Subject code: BTAM101-23
Topic: Sequence
Dr. HemantKumar
Subject code: BTAM101-23
Topic: Sequence
Dr. HemantKumar
Sequence
•Asequenceisanorderedsetofnumbers
�
1,�
2,�
3…
•Here,�
1,�
2,�
3…representthetermsofthe
sequence.Itisdenotedby�
����
�.
•Ifthenumberoftermsisinfinitethenthesequence
issaidtobeinfinitesequenceotherwiseafinite
sequence.
•e.g.,thesetofnumbers2,4,6,8,…isinfinite
sequence,andisrepresentedas2���2�.Here,
�
�=2�isthenthtermofthegeneralterm.
•Asequenceisanorderedsetofnumbers
�
1,�
2,�
3…
•Here,�
1,�
2,�
3…representthetermsofthe
sequence.Itisdenotedby�
����
�.
•Ifthenumberoftermsisinfinitethenthesequence
issaidtobeinfinitesequenceotherwiseafinite
sequence.
•e.g.,thesetofnumbers2,4,6,8,…isinfinite
sequence,andisrepresentedas2���2�.Here,
�
�=2�isthenthtermofthegeneralterm.
•Definition:Asequenceofnumbersisafunction
whosedomainisasetofnaturalnumbers.
�
�:ℕ→ℝ,Here,ℝisthesetofrealnumbers.
Examples:
(i)�
�=(−1)
�
=−1,1,−1,1,…isinfinite
sequence.
(ii)�
�=
�−1
�
=0,
1
2
,
2
3
,…isaninfinitesequence.
(iii)�
�=
1
�
=1,
1
2
,
1
3
,…isaninfinitesequence.
whosedomainisasetofnaturalnumbers.
�
�:ℕ→ℝ,Here,ℝisthesetofrealnumbers.
Examples:
(i)�
�=(−1)
�
=−1,1,−1,1,…isinfinite
sequence.
(ii)�
�=
�−1
�
=0,
1
2
,
2
3
,…isaninfinitesequence.
(iii)�
�=
1
�
=1,
1
2
,
1
3
,…isaninfinitesequence.
Types of sequences
•Convergentsequence:Aconvergentsequenceis
onewhoselimitexists.
i.e.,lim
�→∞
�
�=�(finiteandunique)
•Divergentsequence:Adivergentsequenceisone
whoselimitdoesnotexistor+∞or∞.
i.e.,lim
�→∞
�
�=+∞orlim
�→∞
�
�=−∞
•Oscillatingsequence:Asequencewhichisneither
convergentnordivergent,iscalledoscillating
sequence.
•E.g.,�
�=(−1)
�
=−1,1,−1,1,…
•Convergentsequence:Aconvergentsequenceis
onewhoselimitexists.
i.e.,lim
�→∞
�
�=�(finiteandunique)
•Divergentsequence:Adivergentsequenceisone
whoselimitdoesnotexistor+∞or∞.
i.e.,lim
�→∞
�
�=+∞orlim
�→∞
�
�=−∞
•Oscillatingsequence:Asequencewhichisneither
convergentnordivergent,iscalledoscillating
sequence.
•E.g.,�
�=(−1)
�
=−1,1,−1,1,…
•lim
�→∞
(−1)
�
=ቊ
−1??????��??????����
1??????��??????�����
The above sequence is called oscillatory finite
sequence.
•lim
�→∞
�(−1)
�
=ቊ
−∞??????��??????����
∞??????��??????�����
The above sequence is called oscillatory infinite
sequence.
�→∞
(−1)
�
=ቊ
−1??????��??????����
1??????��??????�����
The above sequence is called oscillatory finite
sequence.
•lim
�→∞
�(−1)
�
=ቊ
−∞??????��??????����
∞??????��??????�����
The above sequence is called oscillatory infinite
sequence.
Convergent of a sequence
•Definition:Thesequence�
�ofnumbersissaid
tobeconvergenttoanumber�ifforevery∈>0,
∃apositiveintegermsuchthat
�
�−�<∈⩝�≥�.
Itiswrittenaslim
�→∞
�
�=�
•Ifthelimitofasequence�
�existsthenthe
sequenceisconvergent.
•Asequence�
�convergestoanumber�thenwe
canwritelim
�→∞
�
�=�or�
�→�.Ifnosuch�
exist,thenwesaythatlimitofsequencedoesnot
existoritisnotconvergent.
•Limitofasequenceitifexiststhenitisunique.
•Definition:Thesequence�
�ofnumbersissaid
tobeconvergenttoanumber�ifforevery∈>0,
∃apositiveintegermsuchthat
�
�−�<∈⩝�≥�.
Itiswrittenaslim
�→∞
�
�=�
•Ifthelimitofasequence�
�existsthenthe
sequenceisconvergent.
•Asequence�
�convergestoanumber�thenwe
canwritelim
�→∞
�
�=�or�
�→�.Ifnosuch�
exist,thenwesaythatlimitofsequencedoesnot
existoritisnotconvergent.
•Limitofasequenceitifexiststhenitisunique.
•Iflim
�→∞
�
�=�,�isfiniteanduniquethensequence
�
�convergesto�,otherwisenot.
•Example:(i)Take�
�=
1
�
thenlim
�→∞
1
�
=0whichis
finiteandunique.Thus,�
�isconvergentand
�
�→0.
(ii)Take�
�=
�
2
+1
�
thenlim
�→∞
�
2
+1
�
→∞
whichisnotfinite.Thus,�
�isnotconvergent.
(iii)�
�=(−1)
�
then
lim
�→∞
(−1)
�
=ቊ
1??????��??????�����
−1??????��??????����
whichisfinitebutnotunique.Thus,�
�isnot
convergent.
�→∞
�
�=�,�isfiniteanduniquethensequence
�
�convergesto�,otherwisenot.
•Example:(i)Take�
�=
1
�
thenlim
�→∞
1
�
=0whichis
finiteandunique.Thus,�
�isconvergentand
�
�→0.
(ii)Take�
�=
�
2
+1
�
thenlim
�→∞
�
2
+1
�
→∞
whichisnotfinite.Thus,�
�isnotconvergent.
(iii)�
�=(−1)
�
then
lim
�→∞
(−1)
�
=ቊ
1??????��??????�����
−1??????��??????����
whichisfinitebutnotunique.Thus,�
�isnot
convergent.
Exercise: Check the convergence of the
sequence �
�where �
�is
(i)
2
??????
−1
3
??????
(ii)
2�+1
1−3�
(iii)
2�
�+1
(iv) (−1)
�+1
2�+3
2�+1
sequence �
�where �
�is
(i)
2
??????
−1
3
??????
(ii)
2�+1
1−3�
(iii)
2�
�+1
(iv) (−1)
�+1
2�+3
2�+1
Sol.
(i)Take �
�=
2
??????
−1
3
??????
lim
�→∞
�
�=lim
�→∞
2
�
−1
3
�
=lim
�→∞
2
3
�
−
1
3
�
=0−0=0
Which is unique & finite. Hence, �
�→0is
convergent sequence.
(ii) lim
�→∞
�
�= lim
�→∞
2�+1
1−3�
=lim
�→∞
2+1/�
1/�−3/�
=
2+0
0−0
→∞
which is not finite. Hence, �
�is not convergent
i.e., divergent.
(i)Take �
�=
2
??????
−1
3
??????
lim
�→∞
�
�=lim
�→∞
2
�
−1
3
�
=lim
�→∞
2
3
�
−
1
3
�
=0−0=0
Which is unique & finite. Hence, �
�→0is
convergent sequence.
(ii) lim
�→∞
�
�= lim
�→∞
2�+1
1−3�
=lim
�→∞
2+1/�
1/�−3/�
=
2+0
0−0
→∞
which is not finite. Hence, �
�is not convergent
i.e., divergent.
(iii) lim
�→∞
�
�=lim
�→∞
2�
�+1
=lim
�→∞
2
1+1/�
=2
Which is unique and finite. Hence, �
�→2is
convergent sequence.
(iv) lim
�→∞
�
�=lim
�→∞
(−1)
�+1
2�+3
2�+1
=
lim
�→∞
(−1)
�+1
2+3/�
2+1/�
=
=ቊ
1??????��??????�����
−1??????��??????����
Since limit is finite but not unique. Hence, �
�is
not convergent.
�→∞
�
�=lim
�→∞
2�
�+1
=lim
�→∞
2
1+1/�
=2
Which is unique and finite. Hence, �
�→2is
convergent sequence.
(iv) lim
�→∞
�
�=lim
�→∞
(−1)
�+1
2�+3
2�+1
=
lim
�→∞
(−1)
�+1
2+3/�
2+1/�
=
=ቊ
1??????��??????�����
−1??????��??????����
Since limit is finite but not unique. Hence, �
�is
not convergent.
(i)
ln�
ln2�
(ii)
3
??????
�
3
(iii)
(−1)
??????+1
2�+1
(iv)
sin�
�
Exercise: Check the convergence of the
sequence �
�where �
�is
ln�
ln2�
(ii)
3
??????
�
3
(iii)
(−1)
??????+1
2�+1
(iv)
sin�
�
Exercise: Check the convergence of the
sequence �
�where �
�is
Sol.
(i)�
�=
ln�
ln2�
⇒ lim
�→∞
�
�=lim
�→∞
ln�
ln2�
=lim
�→∞
1/�
1
2??????
×2
=1(Using
L’Hospitalrule)
Whichisfiniteandunique.Hence,�
�→1isconvergentsequence.
(ii)lim
�→∞
�
�=lim
�→∞
3
??????
�
3
=lim
�→∞
3
??????
ln3
3�
2
=lim
�→∞
3
??????
ln3
2
6�
=
lim
�→∞
3
??????
ln3
3
6
→∞notfinite(UsingL’Hospitalrule)
(iii)lim
�→∞
�
�=lim
�→∞
(−1)
??????+1
2�+1
→0whichisfiniteandunique.
Hence,�
�→0isconvergentsequence.
(iv)lim
�→∞
�
�=lim
�→∞
sin�
�
=lim
�→∞
��.??????�??????�??????�????????????�????????????�−1??????�1
�
→0
whichisuniqueandfinite.
Hence,�
�→0isconvergentsequence.
(i)�
�=
ln�
ln2�
⇒ lim
�→∞
�
�=lim
�→∞
ln�
ln2�
=lim
�→∞
1/�
1
2??????
×2
=1(Using
L’Hospitalrule)
Whichisfiniteandunique.Hence,�
�→1isconvergentsequence.
(ii)lim
�→∞
�
�=lim
�→∞
3
??????
�
3
=lim
�→∞
3
??????
ln3
3�
2
=lim
�→∞
3
??????
ln3
2
6�
=
lim
�→∞
3
??????
ln3
3
6
→∞notfinite(UsingL’Hospitalrule)
(iii)lim
�→∞
�
�=lim
�→∞
(−1)
??????+1
2�+1
→0whichisfiniteandunique.
Hence,�
�→0isconvergentsequence.
(iv)lim
�→∞
�
�=lim
�→∞
sin�
�
=lim
�→∞
��.??????�??????�??????�????????????�????????????�−1??????�1
�
→0
whichisuniqueandfinite.
Hence,�
�→0isconvergentsequence.
Sandwich Theorem For Sequences
•If �
�, �
�, �
�are sequences of real numbers
such that �
�≤�
�≤�
�⩝�and lim
�→∞
�
�=
�;lim
�→∞
�
�=�then lim
�→∞
�
�=�.
•Examples:
(i) �
�=
(−1)
??????+1
2�+1
(ii) �
�=
−1
??????+1
(2�+3)
2�+1
(iii) �
�=
sin�
�
•If �
�, �
�, �
�are sequences of real numbers
such that �
�≤�
�≤�
�⩝�and lim
�→∞
�
�=
�;lim
�→∞
�
�=�then lim
�→∞
�
�=�.
•Examples:
(i) �
�=
(−1)
??????+1
2�+1
(ii) �
�=
−1
??????+1
(2�+3)
2�+1
(iii) �
�=
sin�
�
Sol.
(i)For �
�=
(−1)
??????+1
2�+1
−1
2�+1
≤�
�≤
1
2�+1
⇒−lim
�→∞
1
2�+1
≤lim
�→∞
�
�≤lim
�→∞
1
2�+1
⇒0≤lim
�→∞
�
�≤0
⇒lim
�→∞
�
�=0which is finite and unique and
hence,�
�→0is convergent sequence.
(i)For �
�=
(−1)
??????+1
2�+1
−1
2�+1
≤�
�≤
1
2�+1
⇒−lim
�→∞
1
2�+1
≤lim
�→∞
�
�≤lim
�→∞
1
2�+1
⇒0≤lim
�→∞
�
�≤0
⇒lim
�→∞
�
�=0which is finite and unique and
hence,�
�→0is convergent sequence.
(ii)For�
�=
−1
??????+1
(2�+3)
2�+1
−(2�+3)
2�+1
≤�
�≤
(2�+3)
2�+1
⇒−lim
�→∞
2�+3
2�+1
≤≤lim
�→∞
2�+3
2�+1
⇒−lim
�→∞
2+3/�
2+1/�
≤lim
�→∞
�
�≤lim
�→∞
2+3/�
2+1/�
⇒−1≤lim
�→∞
�
�≤lim
�→∞
1
⇒lim
�→∞
�
�isnotunique.Notconvergent.
�=
−1
??????+1
(2�+3)
2�+1
−(2�+3)
2�+1
≤�
�≤
(2�+3)
2�+1
⇒−lim
�→∞
2�+3
2�+1
≤≤lim
�→∞
2�+3
2�+1
⇒−lim
�→∞
2+3/�
2+1/�
≤lim
�→∞
�
�≤lim
�→∞
2+3/�
2+1/�
⇒−1≤lim
�→∞
�
�≤lim
�→∞
1
⇒lim
�→∞
�
�isnotunique.Notconvergent.
Standard Results: Commonly used limits
•lim
�→∞
ln�
�
=0;
•lim
�→∞
�
1/�
=1;
•lim
�→∞
??????
�
=0;??????<1;
•lim
�→∞
(1+??????/�)
�
=�
�
; ⩝??????;
•lim
�→∞
�
??????
�!
=0;⩝??????
•lim
�→∞
ln�
�
=0;
•lim
�→∞
�
1/�
=1;
•lim
�→∞
??????
�
=0;??????<1;
•lim
�→∞
(1+??????/�)
�
=�
�
; ⩝??????;
•lim
�→∞
�
??????
�!
=0;⩝??????
Exercise: Check the convergence of the
sequence�
�where �
�is
•(i)
�+1
�−1
�
•(ii)
ln�
�
1/??????
•(iii)
�!
�
??????
•(iv) 1−
1
�
2
�
•(v)
4�+1
4�−1
�
sequence�
�where �
�is
•(i)
�+1
�−1
�
•(ii)
ln�
�
1/??????
•(iii)
�!
�
??????
•(iv) 1−
1
�
2
�
•(v)
4�+1
4�−1
�
Monotonic sequence
•Asequence�
�issaidtobemonotonicifitis
eitherincreasingordecreasing.
•Asequence�
�issaidtobemonotonically
increasingif�
�≤�
�+1⩝�∈ℕ
•Asequence�
�issaidtobemonotonically
decreasingif�
�≥�
�+1⩝�
•Theappropriateruletocheckwhether
sequenceisincreasingordecreasing,wemust
check�
�+1−�
�≥0or�
�+1−�
�≤0⩝�
•Asequence�
�issaidtobemonotonicifitis
eitherincreasingordecreasing.
•Asequence�
�issaidtobemonotonically
increasingif�
�≤�
�+1⩝�∈ℕ
•Asequence�
�issaidtobemonotonically
decreasingif�
�≥�
�+1⩝�
•Theappropriateruletocheckwhether
sequenceisincreasingordecreasing,wemust
check�
�+1−�
�≥0or�
�+1−�
�≤0⩝�
Exercise: Check whether the sequence
is monotonic
(i) �
�=
�
�+1
�
�+1−�
�=
�+1
�+2
−
�
�+1
=
�+1
2
−��+2
�+2�+1
=
1
�+2�+1
≥0
Hence, �
�is monotonically increasing
sequence.
is monotonic
(i) �
�=
�
�+1
�
�+1−�
�=
�+1
�+2
−
�
�+1
=
�+1
2
−��+2
�+2�+1
=
1
�+2�+1
≥0
Hence, �
�is monotonically increasing
sequence.
(ii)
3�+1
5�+7
(iii) 1+
1
2
+
1
2
2
+⋯+
1
2
??????
(iv)
−1
3�+5
(v) �
�=1+
1
2
+
1
3
+⋯+
1
�
+
1
�+1
−log�
3�+1
5�+7
(iii) 1+
1
2
+
1
2
2
+⋯+
1
2
??????
(iv)
−1
3�+5
(v) �
�=1+
1
2
+
1
3
+⋯+
1
�
+
1
�+1
−log�
Bounded Sequence
•Asequence�
�issaidtobeboundedabove
if∃arealnumber�suchthat�
�≤�∀�∈ℕ
•Asequence�
�issaidtobeboundedabove
if∃arealnumberℎsuchthatℎ≤�
�∀�∈ℕ
•Asequence�
�iscalledboundedsequenceitis
boundedbelowandboundedabove
i.e.,∃tworealnumbersℎand�suchthat
ℎ≤�
�≤�∀�∈ℕ
•Here,ℎiscalledalowerboundand�iscalled
upperboundofthesequence.
•Asequence�
�issaidtobeboundedabove
if∃arealnumber�suchthat�
�≤�∀�∈ℕ
•Asequence�
�issaidtobeboundedabove
if∃arealnumberℎsuchthatℎ≤�
�∀�∈ℕ
•Asequence�
�iscalledboundedsequenceitis
boundedbelowandboundedabove
i.e.,∃tworealnumbersℎand�suchthat
ℎ≤�
�≤�∀�∈ℕ
•Here,ℎiscalledalowerboundand�iscalled
upperboundofthesequence.
Examples
Example -1: �is bounded below but not bounded above.
Therefore, it is unbounded sequence.
Example-2: −�is bounded above but not bounded below.
Therefore, it is unbounded sequence.
Example-3:
1
�
is bounded sequence as 0≤
1
�
≤1⩝�∈ℕ
Example-4:
(−1)
??????
�
is bounded sequence as −1≤
(−1)
??????
�
≤1
Example-5: �(−1)
�
is an unbounded sequence.
Example -1: �is bounded below but not bounded above.
Therefore, it is unbounded sequence.
Example-2: −�is bounded above but not bounded below.
Therefore, it is unbounded sequence.
Example-3:
1
�
is bounded sequence as 0≤
1
�
≤1⩝�∈ℕ
Example-4:
(−1)
??????
�
is bounded sequence as −1≤
(−1)
??????
�
≤1
Example-5: �(−1)
�
is an unbounded sequence.
Importantpointsaboutabounded
sequence
•Boundedsequenceisalwayswrittenas
I��??????����
�≤�
�≤??????��������
�∀�∈ℕ
Infimum= greatest lower bound
Supremum= least upper bound
•Everyconvergentsequenceisboundedbut
converseneednotbetrue.
•Divergentsequenceisunboundedsequence.
•Finitelyoscillatorysequenceisboundedwhile
Infinitelyoscillatorysequenceisunbounded.
sequence
•Boundedsequenceisalwayswrittenas
I��??????����
�≤�
�≤??????��������
�∀�∈ℕ
Infimum= greatest lower bound
Supremum= least upper bound
•Everyconvergentsequenceisboundedbut
converseneednotbetrue.
•Divergentsequenceisunboundedsequence.
•Finitelyoscillatorysequenceisboundedwhile
Infinitelyoscillatorysequenceisunbounded.
•Monotonicallyincreasingandboundedabove
sequenceisconvergent.
•Monotonicallydecreasingandboundedbelow
sequenceisconvergent.
•Convergentlimitofmonotonicallyincreasing
andboundedabovesequenceissupremumof
thesequence.
•Convergentlimitofmonotonicallydecreasing
andboundedbelowsequenceisinfimumof
thesequence.
sequenceisconvergent.
•Monotonicallydecreasingandboundedbelow
sequenceisconvergent.
•Convergentlimitofmonotonicallyincreasing
andboundedabovesequenceissupremumof
thesequence.
•Convergentlimitofmonotonicallydecreasing
andboundedbelowsequenceisinfimumof
thesequence.
Some important results
•Result-1.
If lim
�→∞
�??????+1
�??????
=�,where �<1,then lim
�→∞
�
�=0
•Result-2.
If lim
�→∞
�??????+1
�??????
=�,where�>1,then lim
�→∞
�
�=+∞
•Result-3.
•For sequence {�
�
} :
(i) diverges to +∞if �>1
(ii) converges if −1<�≤1
(iii) oscillates finitely if �=−1
(iv) oscillates infinitely if �<−1
•Result-1.
If lim
�→∞
�??????+1
�??????
=�,where �<1,then lim
�→∞
�
�=0
•Result-2.
If lim
�→∞
�??????+1
�??????
=�,where�>1,then lim
�→∞
�
�=+∞
•Result-3.
•For sequence {�
�
} :
(i) diverges to +∞if �>1
(ii) converges if −1<�≤1
(iii) oscillates finitely if �=−1
(iv) oscillates infinitely if �<−1
•Cauchy’s First Theorem on Limits
If �
�→�, then ??????
�=
�
1+�
2+�
3+⋯+�
??????
�
→�as �→
∞
•Cauchy’s Second Theorem on Limits
If �
�>0and
�??????+1
�??????
→�, then prove that
??????
�
�→�
If �
�→�, then ??????
�=
�
1+�
2+�
3+⋯+�
??????
�
→�as �→
∞
•Cauchy’s Second Theorem on Limits
If �
�>0and
�??????+1
�??????
→�, then prove that
??????
�
�→�
Important questions
1.Prove that the sequence {�
�}, where
�
�=
1
�+1
+
1
�+2
+
1
�+3
…+
1
2�
is convergent.
Ans. Here, �
�=
1
�+1
+
1
�+2
+
1
�+3
+⋯+
1
2�
∴�
�+1=
1
�+2
+
1
�+3
+
1
�+4
+⋯+
1
2�
+
1
2�+1
+
1
2�+2
∴�
�+1−�
�=
1
2�+1
+
1
2�+2
−
1
�+1
=
1
2�+1
+
1
2(�+1)
−
1
�+1
=
1
2�+1
−
1
2�+2
=
2�+2−2�−1
2�+12�+2
=
1
2�+12�+2
>0⩝�∈ℕ
∴�
�+1−�
�>0⩝�∈ℕ
⇒{�
�}is monotonically increasing
Also �
�=
1
�+1
+
1
�+2
+
1
�+3
…+
1
�+�
<
1
�
+
1
�
+
1
�
…+
1
�
=
�
�
=1⩝�∈ℕ
∴�
�<1⩝�∈ℕ⇒{�
�}is bounded above
∴{�
�}is convergent.
1.Prove that the sequence {�
�}, where
�
�=
1
�+1
+
1
�+2
+
1
�+3
…+
1
2�
is convergent.
Ans. Here, �
�=
1
�+1
+
1
�+2
+
1
�+3
+⋯+
1
2�
∴�
�+1=
1
�+2
+
1
�+3
+
1
�+4
+⋯+
1
2�
+
1
2�+1
+
1
2�+2
∴�
�+1−�
�=
1
2�+1
+
1
2�+2
−
1
�+1
=
1
2�+1
+
1
2(�+1)
−
1
�+1
=
1
2�+1
−
1
2�+2
=
2�+2−2�−1
2�+12�+2
=
1
2�+12�+2
>0⩝�∈ℕ
∴�
�+1−�
�>0⩝�∈ℕ
⇒{�
�}is monotonically increasing
Also �
�=
1
�+1
+
1
�+2
+
1
�+3
…+
1
�+�
<
1
�
+
1
�
+
1
�
…+
1
�
=
�
�
=1⩝�∈ℕ
∴�
�<1⩝�∈ℕ⇒{�
�}is bounded above
∴{�
�}is convergent.
2. Prove that the sequence
2�−7
3�+2
is bounded.
Ans. �
�=
2�−7
3�+2
>0⩝�≥4
Also, �
1=
−5
5
=−1, �
2=
−3
8
, �
3=
−1
11
∴�
�≥−1⩝�
∴{�
�}is bounded below.
Again, �
�=
2
3
−
25
3
3�+2
<2/3⩝�(divide 2n-7 by
3n+2)
∴{�
�}is bounded above.
∴{�
�}is bounded.
2�−7
3�+2
is bounded.
Ans. �
�=
2�−7
3�+2
>0⩝�≥4
Also, �
1=
−5
5
=−1, �
2=
−3
8
, �
3=
−1
11
∴�
�≥−1⩝�
∴{�
�}is bounded below.
Again, �
�=
2
3
−
25
3
3�+2
<2/3⩝�(divide 2n-7 by
3n+2)
∴{�
�}is bounded above.
∴{�
�}is bounded.
3. If �
�=1+
1
2
+
1
3
+⋯+
1
�
−log�,
then prove that �
�is monotonically decreasing sequence. Prove that
it is convergent.
Ans. Here, �
�=1+
1
2
+
1
3
+⋯+
1
�
−log�
∴�
�+1=1+
1
2
+
1
3
+⋯+
1
�
+
1
�+1
−log(�+1)
∴�
�+1−�
�=
1
�+1
−log(�+1)+log�=
1
�+1
−log
�+1
�
=
1
�+1
−log1+
1
�
=
1
�+1
−
1
�
−
1
2�
2
+
1
3�
3
−
1
4�
4
+⋯∵log1+??????=??????−
�
2
2
+
�
3
3
−
�
4
4
+⋯
=
1
�+1
−
1
�
+
1
2�
2
−
1
3�
3
−
1
4�
4
+⋯<
1
�+1
−
1
�
+
1
2�
2
∵
1
3�
3
−
1
4�
4
>0
=
2�
2
−2��+1+�+1
2�
2
(�+1)
=−
�−1
2�
2
(�+1)
≤0⩝�∈ℕ
∴�
�+1−�
�<0⩝�∈ℕ
⇒�
�+1<�
�⩝�∈ℕ
∴�
�is monotonically decreasing. Also �
�> 0, ∴�
�is bounded below.
∴�
�is convergent sequence. Proved.
�=1+
1
2
+
1
3
+⋯+
1
�
−log�,
then prove that �
�is monotonically decreasing sequence. Prove that
it is convergent.
Ans. Here, �
�=1+
1
2
+
1
3
+⋯+
1
�
−log�
∴�
�+1=1+
1
2
+
1
3
+⋯+
1
�
+
1
�+1
−log(�+1)
∴�
�+1−�
�=
1
�+1
−log(�+1)+log�=
1
�+1
−log
�+1
�
=
1
�+1
−log1+
1
�
=
1
�+1
−
1
�
−
1
2�
2
+
1
3�
3
−
1
4�
4
+⋯∵log1+??????=??????−
�
2
2
+
�
3
3
−
�
4
4
+⋯
=
1
�+1
−
1
�
+
1
2�
2
−
1
3�
3
−
1
4�
4
+⋯<
1
�+1
−
1
�
+
1
2�
2
∵
1
3�
3
−
1
4�
4
>0
=
2�
2
−2��+1+�+1
2�
2
(�+1)
=−
�−1
2�
2
(�+1)
≤0⩝�∈ℕ
∴�
�+1−�
�<0⩝�∈ℕ
⇒�
�+1<�
�⩝�∈ℕ
∴�
�is monotonically decreasing. Also �
�> 0, ∴�
�is bounded below.
∴�
�is convergent sequence. Proved.
4. Prove that the sequence 1+
1
�
�
is bounded.
Ans. �
�=1+
1
�
�
, clearly �
�>0⩝�∈ℕ
=1+�
1
�
+
�(�−1)
2!
1
�
2
+
�(�−1)(�−2)
3!
1
�
3
+⋯
=1+1+
(1−1/�)
2!
+
(1−1/�)(1−2/�)
3!
+⋯
<1+
1
1!
+
1
2!
+
1
3!
+⋯<�⩝�∈ℕ
[Since (1-1/n) <1, 1−
2
�
<1]
i.e., �
�<�⩝�∈ℕ
∴0< �
�<�⩝�∈ℕ. Thus, {�
�} is bounded.
1
�
�
is bounded.
Ans. �
�=1+
1
�
�
, clearly �
�>0⩝�∈ℕ
=1+�
1
�
+
�(�−1)
2!
1
�
2
+
�(�−1)(�−2)
3!
1
�
3
+⋯
=1+1+
(1−1/�)
2!
+
(1−1/�)(1−2/�)
3!
+⋯
<1+
1
1!
+
1
2!
+
1
3!
+⋯<�⩝�∈ℕ
[Since (1-1/n) <1, 1−
2
�
<1]
i.e., �
�<�⩝�∈ℕ
∴0< �
�<�⩝�∈ℕ. Thus, {�
�} is bounded.
5. Prove that lim
�→∞
1
�
1+
1
2
+
1
3
+⋯+
1
�
=0
Sol. Let �
�=
1
�
.Then lim
�→∞
1
�
=0
By Cauchy’s first limit theorem on limits, we have
lim
�→∞
�
1+�
2+�
3+⋯+�
�
�
=0
⇒lim
�→∞
1
�
1+
1
2
+
1
3
+⋯+
1
�
=0. Proved
�→∞
1
�
1+
1
2
+
1
3
+⋯+
1
�
=0
Sol. Let �
�=
1
�
.Then lim
�→∞
1
�
=0
By Cauchy’s first limit theorem on limits, we have
lim
�→∞
�
1+�
2+�
3+⋯+�
�
�
=0
⇒lim
�→∞
1
�
1+
1
2
+
1
3
+⋯+
1
�
=0. Proved
6. Prove thatlim
�→∞
2
1
1
3
2
2
4
3
3
…
�+1
�
�
1
??????
=�
Sol. Let �
�=
2
1
1
3
2
2
4
3
3
…
�+1
�
�
�
�+1=
2
1
1
3
2
2
4
3
3
…
�+1
�
�
�+2
�+1
�+1
∴
�
??????+1
�
??????
=
�+2
�+1
�+1
=1+
1
�+1
�+1
⇒lim
�→∞
�
??????+1
�
??????
=lim
�→∞
1+
1
�+1
�+1
=�
Now, �
�>0⩝�∈ℕ
∴by Cauchy’s second theorem on limits, we have
lim
�→∞
�
�
1/�
=�or
lim
�→∞
2
1
1
3
2
2
4
3
3
…
�+1
�
�
1
??????
=�. Proved.
�→∞
2
1
1
3
2
2
4
3
3
…
�+1
�
�
1
??????
=�
Sol. Let �
�=
2
1
1
3
2
2
4
3
3
…
�+1
�
�
�
�+1=
2
1
1
3
2
2
4
3
3
…
�+1
�
�
�+2
�+1
�+1
∴
�
??????+1
�
??????
=
�+2
�+1
�+1
=1+
1
�+1
�+1
⇒lim
�→∞
�
??????+1
�
??????
=lim
�→∞
1+
1
�+1
�+1
=�
Now, �
�>0⩝�∈ℕ
∴by Cauchy’s second theorem on limits, we have
lim
�→∞
�
�
1/�
=�or
lim
�→∞
2
1
1
3
2
2
4
3
3
…
�+1
�
�
1
??????
=�. Proved.
Do yourself.
1. Prove that the sequence
3�−1
4�+5
is
(i) Monotonically increasing
(ii) Bounded
(iii) Convergent
2. Prove that the sequence {�
�}, where
�
�=1+
1
2
+
1
2
2
+⋯+
1
2
??????−1
convergent.
3. Prove that the sequence
�
2
+1
2�+3
is unbounded.
1. Prove that the sequence
3�−1
4�+5
is
(i) Monotonically increasing
(ii) Bounded
(iii) Convergent
2. Prove that the sequence {�
�}, where
�
�=1+
1
2
+
1
2
2
+⋯+
1
2
??????−1
convergent.
3. Prove that the sequence
�
2
+1
2�+3
is unbounded.
4. Show that lim
�→∞
1
�
2
1
+
3
2
+
4
3
+⋯+
�+1
�
=1
5. Show that lim
�→∞
�+1�+2…(�+�)
�
??????
1
??????
=
4
??????
6. Prove that
1
�
2
+
1
(�+1)
2
+
1
(�+2)
2
+⋯+
1
(2�)
2
is bounded sequence.
7. Show that the sequence
�+1
�
converges to 1.
�→∞
1
�
2
1
+
3
2
+
4
3
+⋯+
�+1
�
=1
5. Show that lim
�→∞
�+1�+2…(�+�)
�
??????
1
??????
=
4
??????
6. Prove that
1
�
2
+
1
(�+1)
2
+
1
(�+2)
2
+⋯+
1
(2�)
2
is bounded sequence.
7. Show that the sequence
�+1
�
converges to 1.
ProgramacodeinPythontoidentify
whetherthegivensequenceis
convergentornot.
PROJECT
whetherthegivensequenceis
convergentornot.
PROJECT
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