1. Warm Up Determine the interval of convergence for the series:

2. WARM UP 1.Determine the sum of the infinite geometric series: 2.Which of the following series converge? a) b) c)d)

3. Power SeriesPower Series and elementary functions

4. Consider the series: Write out the first four terms of the series. Does the series converge? How do you know? What is the sum of the series? What if ¼ is replaced by x?

5. The function is called an elementary function and represents the sum of the Power Series

6. Other elementary functions that you must know the Power series for are ln(x) (centered at x = 1) e x (centered at x = 0) cos(x) (centered at x = 0) sin(x) (centered at x = 0) You can determine the power series by using the Taylor polynomial formula until you figure out the pattern. We have already done ln(x) and e x. Determine the Power Series for cos(x) and sin(x).

7. You can use an elementary function Power Series to derive other Power Series Write the first four non-zero terms and the general term for the power series

8. You can use an elementary functions Power Series to derive other Power Series Write the first four non-zero terms and the general term for the power series

9. Power series can be multiplied, divided, added and subtracted like polynomials. Determine the first four nonzero terms and the general term of the series: xsin(x)

10. Determine the first four nonzero terms and the general term of the series:

12. You can verify derivatives using Power Series Use Power series to show that the derivative of sin (x) is cos (x)

13. Given the derivative of arctan(x) is Determine the first four nonzero terms and the general term of the Taylor polynomial about x = 0 for arctan(x)