Radhika Uniyal - Dissertation PPT (1).pptx

This is an PPT On Algebra

1. DISSERTATION THEME – SERIES OF FUNCTIONS Presented By – Radhika Uniyal Supervisor – B.Sc Vth Semester Mrs Richa Dhiman

2. Sequence of real numbers A real sequence is a function f : N R < f(1), f(2), f(3), f(4), . . . . . . . . . f(n), f(n+1) . . . . . > or < a₁, a₂, a₃, a₄, . . . . . . a n, a n+1 . . . . . > or < a n > ; < f(n) > For example :- f(n) = h ; < f(n) > = < n > < 1, 2, 3, 4 . . . . >

3. Range Of a Sequence Let < a n > be a real sequence, then range of < a n > define by { f(x) : x є N } Ex :- < (-1) n > = < -1, 1, -1, 1 . . . . . . > Range of <(-1) n > = {-1, 1}

4. Types of Sequence Constant Sequence :- If f(n) = Constant then <f(n)> is a constant sequence. EX :- <1, 1, 1, . . . . > = < 1 > Bounded Sequence or f is bounded V n є N . < 1/n > = < 1, ½, 1/3, . . . . > ―> bounded between 1 & 0. Monotonically increasing Sequence :- <f(n)> is said to be monotonically increasing sequence if f(n) <= f(n+1) V n є N. EX :- <n> = <1, 2, 3, 4, 5, 6, . . . . >

5. Types of Sequence…….Contd. Strictly increasing sequence :- <f(n)> is said to be strictly increasing sequence if f(n) < f(n+1) V n є N. EX :- <n 2 > = < 1, 2 2 , 3 2 , 4 2 , . . . .> = < 1, 4, 9, 16, . . . .> Monotonically decreasing sequence :- <f(n)> is said to be monotonically decreasing sequence if ; f(n) >= f(n+1) V n є N. EX :- <-n> = < -1, -2, -3, -4, -5, . . . . > Strictly decreasing sequence :- <f(n)> is said to be strictly decreasing sequence if f (n) > f (n+1) V n є N. EX :- <1/n> = < 1, ½ , 1/3, . . . . . >

6. Limit Point of a Sequence Let <a n > be a real sequence then a real number ‘ l ’ is said to be a limit point of a sequence <a n > if every nbd of l contains infinite many terms of a sequence <a n >. OR for any E > 0 ; a n E(l – E, l + E) for infinitely many values of n. EX :- < 1, 2, 1, 2, . . . . > ―> Limit Point :- { 1, 2 }

7. Cauchy Sequence A real sequence <a n > is said to be cauchy sequence if for any E > 0 there exist n є N such that l a n – a m l < E V n , m >= N. OR l a n – a m l < V n > m > N : N є l N Result :- A real sequence < x n > is cauchy iff it is Convergent. EX :- 1. < 1/n > → Convergent ↔ Cauchy. EX :- 2. < 1 + 1/n > →Convergent ↔ Cauchy.

8. Divergent Sequence Divergent to +∞ :- if for any real no. K € l R ; there exist m є l N such that : a n є [k, ∞) V n >= m or a n >= K V n >= m EX :- <n> = < 1, 2, 3, 4, 5, 6, 7, . . . . > divergent to +∞ Divergent to -∞ :- if for any real no. K є l R ; there exist m є l N such that : a n є [k, ∞ ) V n >= m or a n <= K V n <= m EX :- <-n> = <-1, -2 2 , -3 2 , . . . . > divergent to -∞

9. Oscillatory Sequence A sequence which is neither divergent nor convergent. EX :- 1. < (-1) n > is oscillating sequence. 2. < 1, 2, 1, 2, 1, 2, . . . . > is oscillating sequence. Oscillating finitely :- Oscillating + bounded EX :- < (-1) n > = < -1, 1, -1, 1, . . . . . > Oscillating infinitely :- Oscillating + unbounded EX :- < 1, 2, 1, 3, 1, 4, 1, 5, . . . . . >

10. SERIES Let <a n > be a sequence of real no. then an expression of the form a n is called a series of real numbers. EX :- ∑n = 1 + 2 + 3 + . . . . + n + . . . . Sequence of partial sums :- Let ∑a n be a series of real numbers, then S n = a 1 + a 2 + a 3 + . . . . . + a n S 1 = a 1 , S 2 = a 1 + a 2 , S 3 = a 1 + a 2 + a 3 then, < S n > = < S 1, S 2, S 3, . . . > is said to be sequence of partial sums.

11. Series & Sequence Of Function :- An expression of the form f n (x) ; where f n are functions defined on common domain A, is said to be a Series of function. EX :- Let f n (x) = (cos nx)/n 2  defined on [0, 1] then the series of functions is given by : f n (x) = f 1 (x) + f 2 (x) + f 3 (x) + . . . . . . A sequence of the form <f n (x)> ; where f n (x) are function defined on common domain D, is said to be a Sequence of function. EX :- < x n > = < x 1 , x 2 , x 3 , x 4 , . . . . >

12. Continuity & Uniform Continuity Continuity Uniform continuity Define at a point S ( E, a), S depends on E & a. Continuity ≠ Uniform Continuity . EX :- 1/x on (0, 1) Continuous function but not uniform continuous. Define on a set S depends on E. Uniform Continuity = Continuity

13. Lebinitz Test :- Let <a n > e a monotonically decreasing sequence of +ve real numbers such that a n = 0 then, series (-1) n a n is convergent. EX :- (-1) n /n is convergent ; 1/n = 0 Absolute Convergent :- A series ∑a n is said to be absolute convergent if ∑ la n l is convergent. EX :- ∑(-1) n /n 2 is absolute convergent.

14. Conditional Convergent If a series ∑a n is conditional convergent then ∑ la n l is not convergent. Ex :- ∑ (-1 n )/n is convergent by Lebinitz test by ∑ l (-1 n )/n l = ∑ 1/n is convergent by p – test implies that ∑ (-1 n )/n is conditional convergent. POWER SERIES :- A series of the form a n (x - a) n is said to be a Power Series with center x = a if a = 0 ∑a n x n . EX :- ∑(1/n)x n , ∑(1/n!)x n , ∑(1/n 2 )x n etc.

15. Cauchy Condensation Test Let <a n > be monotonically decreasing sequence of non-negative real numbers, then the series ∑a n is convergent iff ∑2 n a 2n is convergent. EX :- 1/n(log n) 2 = convergent Comparison Test :- ∑a n & ∑b n be two positive term series of real numbers, then a n <= k.b n V n >= m ; m > N ; N є l N then ; a n <= kb n ∑b n < ∞ → ∑a n is convergent. a n /b n = 0 then, a n < b n ∑ b n < ∞ → ∑a n < ∞ ∑ b n is convergent → ∑a n is convergent.

16. Ratio Test Root Test Let ∑a n be a positive term series of real numbers such that ; a n +1/a n = l l < 1 → ∑a n < ∞ [Convergent] l > 1 → ∑a n > ∞ [Divergent] l = 1 → test fail NOTE :- Root test is stronger than Ratio test. Let ∑a n be a positive term series of real numbers such that ; ( a n ) 1/n = l l < 1 → ∑a n < ∞ [Convergent] l > 1 → ∑a n > ∞ [Divergent] l = 1 → test fail

17. Fourier Series Fourier series is an infinite series representation of periodic function in terms of trigonometry function sine and cosine. f(x) = a + (a n cosnx + b n sinnx); a < x < a + 2 π It is known as Fourier – euler formula, where a = 1/2 π f(x)dx a n = 1/ π f(x) cosnxdx b n = 1/ π f(x) sinnxdx

18. Periodic functions If the value of each ordinate f(t) repeats itself at equal intervals in the abscissa, then f(t) is said to be periodic function. If f(t) = f ( t + T ) = f ( t + 2T ) . . . . . Then T is called the period of the function f(t). For example :- The period of sin x , cos x , sec x and cosec x is 2 π . The period of tan x and cot x is π .

19. Example :- Find the fourier series representing, f(x) = x ; 0 < x < 2 π and sketch its graph from x = -4 π to x = 4 π Sol :- Let f(x) = a /2 + a 1 cosx + a 2 cos2x + . . . . + b 1 sin2x + . . . . Hence a = 1/ π f(x)dx = 1/ π xdx = 2 π a n = 1/ π f(x) cosnx dx = 1/ π x cosnx dx = 1/ π [ ( xsinnx/n ) – 1 . ( -cosnx/n 2 ) ] 2 π = 1/ π [ ( cos2n π /n 2 ) – 1/n 2 ] = ( 1/n 2 π ) ( 1-1 ) = 0 Continued . . . .

20. b n = 1/ π f(x) sinx dx = 1/ π x sinnx dx = 1/ π [ x (- cosnx/n)-1 . (-sinnx/n 2 )] 2 π = 1/ π [ -2 π cos2n π /n ] = -2/n Substituting the values of a , a 1 , a 2 . . . b 1 , b 2 , . . . . In 1, we get ; x = π – 2 [ sinx + ½ sin2x + 1/3 sin3x + . . . . ] f(t) - 4 π - 2 π 2 π 4 π

21. Thank You