Introduction

Welcome to this guide on Alternating Series Test for Convergence, this article will guide you through the following:

Previously, we were working with series that only had positive numbers. Now, we will be dealing with series that alternate between positive and negative numbers.

Alternating Series

We define alternating series as a series where the sign of terms switch from positive to negative, or negative to positive.

Eg: $\sum _{n=0}^{\infty }(-1{)}^n{a}_n\text{ and }\sum _{n=1}^{\infty }\cos (n\pi )a_n$

Note that $\cos (n\pi )$ is equivalent to $(-1{)}^n$, given that $n$ is an integer.

Alternating Series Test for Convergence

Let $a_n>0$, the alternating series:

$\sum _{n=1}^{\infty }(-1{)}^n{a}_n\text{ and }\sum _{n=1}^{\infty }(-1{)}^{n+1}{a}_n$

converge if the following two requirements are met:

Do not use this test to test for divergence

Example 1:

Determine the convergence of this series:

$\sum _{n=1}^{\infty }(-1{)}^{n+1}\frac{1}{n}$

This series is known as the Alternating Harmonic Series, to check its convergence, first check the requirements of the test:

Both requirements fit, meaning that this series converges, although this series looks like the harmonic series. 

Practice

Determine if the following series converge using the comparison test: