19. Section 10.4 Absolute and Conditional Convergence

Section 10.4 Absolute and Conditional Convergence EQ – What do you do if the series is not strictly positive?

Absolute and Conditional Convergence So far we have only looked at tests to use if series have all positive terms Today we will add a test for alternating terms (alternate between positive and negative) These take the form c0 - c1+ c2 - c3+c4 . . . Or -c0+c1- c2 + c3 - c4 . . .

One of the most important Examples is: The Alternating Harmonic Series The alternating harmonic series seems to converge to a point about here In order to determine whether the series converges, we need to examine the partial sums of the series. 1 2 3 4 5 6 7 8 9 10 1/2 1

Alternating Series Test (also known as Leibniz Theorem) Converges if: an an+1 for all n  N for some integer N. The AST can ONLY be used to prove convergence. If any condition is not met, you cannot say anything about series using this test.

The Idea behind the AST We have already seen the crucial picture. a0 S 1 2 3 4 5 6 7 8 9 10 a0 S a0-a1 There were two things that made this picture “go.” The size (in absolute value) of the terms was decreasing. The terms were going to zero.

Example The first condition isn’t met and so there is no reason to check the second. Since this condition isn’t met we’ll need to use another test to check convergence. In these cases where the first condition isn’t met it is usually best to use the divergence test. Diverges

Example Converges

Example Alternating Harmonic Converges

Alternating Series Remainder If an alternating series satisfies the conditions of the AST, you can approximate the sum of the series by using the nth partial sum, Sn, and the error will have an absolute value of NO GREATER THAN the first term left off, an+1 This means that the interval in which S (the sum) must lie is [Sn - Rn , Sn + Rn ] where R is the error

Example Approximate the sum by using its first six terms, and find the error. Then write the interval in which S must lie.

Example Approximate the sum with an error of less than 0.001.

Absolute and Conditional Convergence A series is absolutely convergent if the corresponding series of absolute values converges. A series that converges but does not converge absolutely, converges conditionally. Every absolutely convergent series converges. (Converse is false!!!)

This is not an alternating series, but since Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent? This is not an alternating series, but since Is a convergent geometric series, then the given series is absolutely convergent.

Converges by the Alternating series test. b) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent? Converges by the Alternating series test. Diverges with direct comparison with the harmonic series. The given series is conditionally convergent.

c) Is the given series convergent or divergent c) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent? By the nth term test for divergence, the series diverges.

Converges by the alternating series test. d) Is the given series convergent or divergent? If it is convergent, its it absolutely convergent or conditionally convergent? Converges by the alternating series test. Diverges since it is a p-series with p <1. The Given series is conditionally convergent.

Assignment Pg. 588: #1-27 odd