# 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:

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 *a _{k}* = 1/

*k*satisfy:

*a*<

_{k+1}*a*. Moreover, the terms in the sequence alternate between positive and negative.

_{k}Therefore the alternating series test can be applied.

### Step 2: Apply the Alternating Series Test

The series is therefore convergent.

## Explanation of Each Step

### Step (1)

Given a general alternating series,

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

- terms in the series must alternate between positive and negative
- the terms in the sequence must be decreasing (in other words,
*a*)_{k}< a_{k-1}

Our series meets these two criteria.

### Step (2)

The alternating series test on the general alternating series

only requires that we evaluate

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