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:

8. Solution

9. Example(2)

10. We notice that And we know that:

11. Solution

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

13. 1. The function ln(x-1) 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

14. Approximating ln2

16. Example(3)

17. We notice that And we know that:

19. Solution

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. Approximating arctan (0.5)

25. Showing that this series converges to e

27. Approximating e

28. Question Approximate 3 √e

29. Able’s Theorem

31. Home Quiz (2)

32. Homework