Sequence and Series of Functions

1. Sequence and Series of Functions

2. Sequence of functions • Definition: A sequence of functions is simply a set of functionsun(x), n = 1, 2, . . . defined on a common domain D. • A frequently usedexample will be the sequence of functions {1, x, x2, . . .}, x ϵ [-1, 1]

3. Sequence of Functions Convergence • Let D be a subset of  and let {un} be a sequence of real valued functions defined on D. Then {un} converges on D to g if for each x ϵD • 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

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

5. 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 

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

7. Sequence of Functions Convergence • Example 3: Consider the sequence {un} of functions defined by Show that {un} converges for all x in 

8. Sequence of Functions Convergence • Solution For every real number x, we have Thus, {un} converges to the zero function on 

9. Sequence of Functions Convergence • Example 4: Consider the sequence {un} of functions defined by Show that {un} converges for all x in 

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

11. 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

12. Sequence of Functions Convergence • Example 6 Consider the sequence {fn} 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 

13. Sequence of Functions Convergence

14. Sequence of Functions Convergence

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

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

17. Series of Functions • Definition: Aninfinite series of functions is given by x ϵD.

18. Series of Functions Convergence • is said to be convergent on D if the sequence of partial sums {Sn(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

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

20. Series of Functions Convergence • Example 8: Find the domain of convergence of (1 – x) + x(1 – x) + x2(1 – x) + ....

21. Series of Functions Convergence • Example 9: Investigate the uniform convergence of

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

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

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