Return to Our Motivating Example

Example

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

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

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

Therefore the alternating series test can be applied.

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The series is therefore convergent.

Given a general alternating series,

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

Our series meets these two criteria.

The alternating series test on the general alternating series

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only requires that we evaluate

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