First, in ratio test you'll calculate the limit of the fraction as k goes to infinity not just the fraction!
Second, the ratio test will be failed.
Third, I'll use the comparison test for the absolute series.
Determine if the following alternating series converges. Is it absolutely convergent?
Using the alternating series test, we need:
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.