2. Consider the sentence For what values of x is this an identity? On the left is a function with domain of all real numbers, and on the right is a limit of Taylor polynomials… As we have already explored in previous sections (check the graph!), these polynomials only converge to the function over the interval (–1, 1)… While the graphs we have used provide compelling visual support, they don’t actually prove convergence…

3. The Convergence Theorem for Power Series There are three possibilities for with respect to convergence: 1. There is a positive number R such that the series diverges for but converges for. The series may or may not converge at either of the endpoints and. 2. The series converges for every x. 3. The series converges at x = a and diverges elsewhere

4. The Convergence Theorem for Power Series The number R is the radius of convergence, and the set of all values of x for which the series converges is the interval of convergence. The radius of convergence completely determines the interval of convergence if R is either zero or infinite. For all other R values, there remains the question of what happens at the endpoints of the interval (recall that the table on p.477 includes intervals of convergence that are open, half-open, and closed). The endpoint question will be addressed in the final section of this chapter…

5. The nth-Term Test for Divergence diverges if fails to exist or is not zero. Essentially, a convergent series must have the nth term go to zero as n approaches infinity.

6. The Direct Comparison Test Let be a series with no negative terms. We can check a series for convergence by comparing it term by term with another known convergent or divergent series. (a) converges if there is a convergent series with for all, for some integer. (b) diverges if there is a divergent series of nonnegative terms with for all, for some integer.

7. The Ratio Test Let be a series with positive terms, and with Then, (a) the series converges if, (b) the series diverges if, (c) the test is inconclusive if.

8. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. nth-Term Test: Because the “final” term fails to exist, the series diverges.

9. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. Ratio Test: Because this ratio is greater than one, the series diverges.

10. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. This is simply a geometric series with Because, the series converges.

11. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. nth-Term Test: The series diverges. Let

12. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. Ratio Test: The series converges.

13. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. The series diverges. nth-Term Test:

14. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. The series diverges. Ratio Test:

15. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. Ratio Test:

16. Guided Practice For each of the following, determine the convergence or divergence of the series. Identify the test (or tests) you use. There may be more than one correct way to determine convergence or divergence of a given series. The series converges.