1. Representing Functions by Power Series

2. A power series is said to represent a function f with a domain equal to the interval I of convergence of the series if the series converges to f(x) on that interval. That’s if:

3. Example

4. Theorem

5. Examples

6. Example(1)

7. We notice that And we know that:

9. Detailed Explanation

10. Example(2)

11. We notice that And we know that:

12. Solution of Example (2)

13. Detailed Explanation

14. Question What about the convergence at the end points?

15. 1. The function ln(1-x) is not defined at x = 1 2. We can show easily that the series is convergent if x = -1 (how?) But does it converge to ln2? The answer to this question has to wait till we introduce Able’s Theorem

16. Example(3)

17. We notice that And we know that:

20. Question What about the convergence at the end points?

21. We can show easily that the series is convergent if x = 1or x = -1 (how?) But does it converge to arctan1 = π/4 & arctan(-1) = π/4 respectively ? The answer to this question has to wait until after we introduce Able’s Theorem !

22. Example(4)

23. Solution

24. Example(5)

25. Solution

26. Example(6)

27. Solution

28. Example(7)

29. Solution

30. Example(8)

31. Solution

32. Example(9)

33. Solution

35. Showing that this series converges to e

37. Able’s Theorem

39. Assignment