Absolute vs. Conditional Convergence Alternating Series and the Alternating Series Test

Series with Positive Terms Recall that series in which all the terms are positive have an especially simple structure when it comes to convergence. Because each term that is added is positive, the sequence of partial sums is increasing. So one of two things happens: 1. The partial sums stay bounded and the series converges, 2. The partial sums go off to infinity and the series diverges. OR

What happens when a Series has some terms that are negative? All the terms are negative. Infinitely many negative terms and infinitely many positive terms. Finitely many terms are negative. There are several Possibilities:

Alternating Series Definition: An alternating series is one whose terms alternate in sign. For a sequence (c n ) of positive numbers, there are two possibilities: c 0 - c 1 + c 2 - c 3 +c 4... Or -c 0 +c 1 - c 2 + c 3 - c 4... In some ways, this situation is the most conducive to convergence, since the positive and negative terms have a tendency to cancel each other out, thus preventing the partial sums from getting too large. Note: the N th term test for divergence still applies. Consider:

One of the most important Examples is: The Alternating Harmonic Series In order to determine whether the series converges, we need to examine the partial sums of the series. Look at Example 1 on pg. 576 of OZ. The alternating harmonic series seems to converge to a point about here /2 1

This suggests The Alternating Series Test Theorem: (Alternating Series Test) Consider the series c 1 - c 2 + c 3 - c 4... and -c 1 + c 2 - c 3 + c 4... Where c 1 > c 2 > c 3 > c 4 >...> 0 and Then the series converge, and each sum S lies between any two successive partial sums. That is, for all k, either or depending on whether k is even or odd.

The Idea behind the AST We have already seen the crucial picture. a0-a1a0-a a0a0 S The error estimate given at the end of the theorem is also obvious from the picture. Note:

The Idea behind the AST We have already seen the crucial picture. a0-a1a0-a a0a0 S 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.

Absolute and Conditional Convergence Definition: Let be any series If converges, then is said to converge absolutely. If diverges but converges, then is said to converge conditionally. Fact: Any series that converges absolutely, also converges in the partial sum sense. In this case, the absolute sum of the series is greater than or equal to the sum of the series. The converse is not true. It is possible for a series to converge but not to converge absolutely. Quintessential example: the alternating harmonic series.

The Alternating Series Test (revised) Theorem: (Alternating Series Test) Consider the series c 1 - c 2 + c 3 - c 4... and -c 1 +c 2 - c 3 + c 4... Where c 1 > c 2 > c 3 > c 4 >...> 0 and Then the series converge, and each sum S lies between any two successive partial sums. In particular, for all k, the error in approximation is