# Science:Infinite Series Module/Units/Unit 2/2.3 The Alternating Series Test/2.3.04 An Alternating Series Example

## Example

We now return to the example we presented at the beginning of the lesson:

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}}$

We wish to determine whether it is convergent using the alternating series test.

## Complete Solution

### Step 1: Check to see if the alternating series test can be applied

We see that the terms ak = 1/k satisfy: ak+1 < ak. Moreover, the terms in the sequence alternate between positive and negative.

Therefore the alternating series test can be applied.

### Step 2: Apply the Alternating Series Test

${\displaystyle \lim _{k\rightarrow \infty }{\frac {1}{k}}=0}$

The series is therefore convergent.

## Explanation of Each Step

### Step (1)

Given a general alternating series,

${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}a_{k}}$

we need only two criteria to apply the alternating series test for a given infinite series:

1. terms in the series must alternate between positive and negative
2. the terms in the sequence must be decreasing (in other words, ak < ak-1)

Our series meets these two criteria.

### Step (2)

The alternating series test on the general alternating series

${\displaystyle \sum _{k=1}^{\infty }(-1)^{k}a_{k}}$

only requires that we evaluate

${\displaystyle \lim _{k\rightarrow \infty }a_{k}}$

If the limit is zero, the series is convergent. In this case, the limit is zero, so our sequence is convergent.