Calculus Applets

Alternating Series Test

The alternating series test is used when the terms of the underlying sequence alternate. Suppose we have a series where the a_n alternate positive and negative. If a_n+1 < a_n (i.e., the terms get smaller) and if then the series converges. If a series Σ | a_n | converges then the series Σ a_n converges and is said to converge absolutely. If Σ | a_n | diverges but Σ a_n converges, then Σ a_n is said to converge conditionally.

Try the following:

1. The applet shows the series , called the alternating harmonic series because its terms alternate sign: . The harmonic series diverges, but maybe the minus signs change the behavior in this case. The alternating series test requires that the a_n alternate sign, get smaller and approach zero as n approaches infinity, which is true in this case. So this series does converge and is said to converge conditionally. While the harmonic series diverges, the alternating harmonic series has enough negative numbers in it to counterbalance the growth, resulting in convergence.

This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.