Testing Convergence at Endpoints

1. Testing Convergenceat Endpoints Section 9.5b

2. Alternating Series A series in which the terms are alternatingly positive and negative is an alternating series. Examples: This is a geometric series with a = –2 and r = –1/2 So it converges to:

3. Alternating Series A series in which the terms are alternatingly positive and negative is an alternating series. Examples: This series diverges by the nth-Term Test: This is the alternating harmonic series, and it converges by a new test that we will see shortly…

4. The Alternating Series Test (Leibniz’s Thm) The series converges if all three of the following conditions are satisfied: 1. each is positive; 2. for all , for some integer ; 3. . The Alternating Series Estimation Thm If the alternating series satisfies the conditions of Leibniz’s Theorem, then the truncation error for the nth partial sum is less than and has the same sign as the first unused term.

5. The Alternating Series Test (Leibniz’s Thm) Closing in on the sum of a convergent alternating series: This figure not only proves the fact of convergence; it also shows the way that an alternating series converges. The partial sums keep “overshooting” the limit as they go back and forth on the number line, gradually closing in as the terms tend to zero.

6. The Alternating Series Test (Leibniz’s Thm) Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms. The terms are strictly alternating in sign and decrease in absolute value from the start: Also, converges. By the Alternating Series Test,

7. The Alternating Series Test (Leibniz’s Thm) Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms. However, the series of absolute values is the harmonic series, which we know diverges, so the alternating harmonic series is not absolutely convergent. The Alternating Series Estimation Theorem guarantees that the truncation error after 99 terms is less than

8. Absolute and Conditional Convergence Because the alternating harmonic series is convergent but not absolutely convergent, we say it is conditionally convergent (or converges conditionally). We take it for granted that we can rearrange the term of a finite sum without affecting the sum. We can also rearrange a finite number of terms of an infinite series without affecting the sum. But if we rearrange an infinite number of terms of an infinite series, we can be sure of leaving the sum unaltered only if it converges absolutely.

9. Practice Problems Determine whether the given series converges absolutely, converges conditionally, or diverges. Give reasons for your answer. is a decreasing series with positive terms, and So the series converges by the Alternating Series Test. Check for absolute convergence:

10. Practice Problems Determine whether the given series converges absolutely, converges conditionally, or diverges. Give reasons for your answer. behaves like for large values of n… diverges, so also diverges… Conclusion: The original series converges conditionally

11. Practice Problems Determine whether the given series converges absolutely, converges conditionally, or diverges. Give reasons for your answer. Take a look at: is always less than or equal to: So this series also converges by direct comparison Which we know converges as a p-series with p = 2. Conclusion: The original series converges absolutely

12. Intervals of Convergence How to Test a Power Series for Convergence 1. Use the Ratio Test to find the values of x for which the series converges absolutely. Ordinarily, this is an open interval In some instances, the series converges for all values of x. In rare cases, the series converges only at x = a. 2. If the interval of convergence is finite, test for convergence or divergence at each endpoint. The Ratio Test fails at these points. Use a comparison test, the Integral Test, or the Alternating Series Test.

13. Intervals of Convergence How to Test a Power Series for Convergence 3. If the interval of convergence is , conclude that the series diverges (it does not even converge conditionally) for , because for those values of x the nth term does not approach zero. Finally, take a look at the flowchart on p.505, and read the “Word of Caution”…

14. Practice Problems Find (a) the interval of convergence of the series. For what values of x does the series converge (b) absolutely, (c) conditionally? This is a geometric series which converges only for (a) (b) (c) None

15. Practice Problems Find (a) the interval of convergence of the series. For what values of x does the series converge (b) absolutely, (c) conditionally? Ratio Test for absolute convergence: The series converges absolutely when Now, we check the endpoints…

16. Practice Problems Find (a) the interval of convergence of the series. For what values of x does the series converge (b) absolutely, (c) conditionally? Check converges conditionally diverges Check (b) (c) At (a)

17. Practice Problems Find (a) the interval of convergence of the series. For what values of x does the series converge (b) absolutely, (c) conditionally? Ratio Test for absolute convergence: This series converges absolutely for all real numbers. (a) All real numbers (b) All real numbers (c) None