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.
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.
There is a ratio test without calculating the limit:
If the series of general positive term satisfies for sufficiently large and fixed then , the series is convergent. In our case does not exist such .
Fernando Revilla
Dirichlet's Test -- from Wolfram MathWorld
Kind regards
Better:
and
is divergent.
Fernando Revilla