Sequence and Series of Functions

Sequence of functions Definition: A sequence of functions is simply a set of functions u n (x), n = 1, 2,... defined on a common domain D. A frequently used example will be the sequence of functions {1, x, x 2,...}, x ϵ [-1, 1]

Sequence of Functions Convergence Let D be a subset of  and let {u n } be a sequence of real valued functions defined on D. Then {u n } converges on D to g if for each x ϵ D More formally, we write that if given any x ϵ D and given any  > 0, there exists a natural number N = N(x,  ) such that

Sequence of Functions Convergence Example 1 Let {u n } be the sequence of functions on  defined by u n (x) = nx. This sequence does not converge on  because for any x > 0

Sequence of Functions Convergence Example 2: Consider the sequence of functions The limits depends on the value of x We consider two cases, x = 0 and x  0 1. x = 0  2. x  0 

Sequence of Functions Convergence Therefore, we can say that {u n } converges to g for |x| < , where

Sequence of Functions Convergence Example 3: Consider the sequence {u n } of functions defined by Show that {u n } converges for all x in 

Sequence of Functions Convergence Solution For every real number x, we have Thus, {u n } converges to the zero function on 

Sequence of Functions Convergence Example 4: Consider the sequence {u n } of functions defined by Show that {u n } converges for all x in 

Sequence of Functions Convergence Solution For every real number x, we have Moreover, Applying the squeeze theorem, we obtain that Therefore, {u n } converges to the zero function on 

Sequence of Functions Convergence Example 5: Periksalah kekonvergenan barisan fungsi pada himpunan bilangan real Solution: Akan ditinjau untuk beberapa kasus: 1. |x| < 1  2. |x| > 1  tidak ada 3. x = 1  4. x = -1  tidak ada Barisan tersebut konvergen untuk  1 < x ≤ 1

Sequence of Functions Convergence Example 6 Consider the sequence {f n } of functions defined by We recall that the definition for convergence suggests that for each x we seek an N such that. This is not at first easy to see. So, we will provide some simple examples showing how N can depend on both x and 

Sequence of Functions Convergence

Uniform Convergence Let D be a subset of  and let {u n } be a sequence of real valued functions defined on D. Then {u n } converges uniformly on D to g if given any  > 0, there exists a natural number N = N(  ) such that

Uniform Convergence Example 7: Ujilah konvergensi uniform dari example 5 a. pada interval -½ < x < ½ b. pada interval -1 < x < 1

Series of Functions Definition: An infinite series of functions is given by x ϵ D.

Series of Functions Convergence is said to be convergent on D if the sequence of partial sums {S n (x)}, n = 1, 2,...., where is convergent on D In such case we write and call S(x) the sum of the series More formally, if given any x ϵ D and given any  > 0, there exists a natural number N = N(x,  ) such that

Series of Functions Convergence If N depends only on  and not on x, the series is called uniformly convergent on D.

Series of Functions Convergence Example 8: Find the domain of convergence of (1 – x) + x(1 – x) + x 2 (1 – x) +....

Series of Functions Convergence Example 9: Investigate the uniform convergence of

Exercise 1.Consider the sequence {f n } of functions defined by for 0 ≤ x ≤ 1. Determine whether {f n } is convergent. 2.Let {f n } be the sequence of functions defined by for  /2 ≤ x ≤  /2. Determine the convergence of the sequence. 3.Consider the sequence {f n } of functions defined by on [0, 1] Show that {f n } converges to the zero function

Exercise 4. Find the domain of convergence of the series a) b) c) d) e) 5. Prove that converges for -1 ≤ x < 1

Exercise 6. Investigate the uniform convergence of the series 7. Let Prove that {f n } converges but not uniformly on (0, 1)