This is a question about convergent and divergent series. Please read everything below before responding.?

Question

The series (Sigma where k=1 up to infinity) [(-1)^k] / ln(k) either diverges, converges absolutely or converges conditionally. I believe this series converges absolutely b/c it's easy to see that the absolute value of this summation will converge, thus eliminating the prospect of conditional convergence and divergence. However, I am not sure how to test if an alternating series converges or diverges. In case this is not clear enough, please refer to this link where this problem is posted. It is in the bottom right corner of the page, Problem sc.

http://www.math.ufl.edu/~squash/Cl/a-cl-prereq-GEN.pdf

Additionally, this is not a homework problem, just a review problem for a test.

Answer 1

First off, I'm assuming that you have a typo in the statement of the series, since the first term of your series would have us divide by 0.

Even once that typo is fixed, however, the series doesn't converge absolutely.  There are many ways to show this.  The easiest for me is comparison to 1/k, which diverges.  Since 1 / ln(k) > 1/k as k goes to infinity, our original series can't converge absolutely.

I believe that an alternating series will at least converge conditionally when the limit of the terms is zero.  This is the case here, so it converges conditionally.

Answer 2

As the previous poster has pointed out, the index k should start with, say, 2; (-1)^k / ln(k) at k=1 involves division by zero, which is not defined. I think this was just a slip, not a trick question.

Given that, the Alternating Series test says that if the terms are decreasing in absolute value, and the limit on the kth term as k→∞ is zero, then the series converges.

And as has already been stated, the series does not converge absolutely, by the comparison test using the known divergent series ∑(1/k).

Therefore, this series converges conditionally.