The question:

Determine if the following alternating series converges. Is it absolutely convergent?

My attempt:

Using the alternating series test, we need:

a)

b)

c)

a) > 0 for all k > 1

b) for all k > 1

c) Limit is obviously 0 as

Thus the series converges.

Now we consider

Using ratio test:

=

Therefore converges. So this series should be absolutely convergent. However the answer is actually 'conditionally convergent'.

Where have I gone wrong? Thanks.