A student was recently asked this question by his instructor:
Converge or diverge?
I feel a little dumb for not being able to answer it. The following tests fail to prove convergence or divergence:
nth term test for divergence (limit is 0)
ratio test (limit is 1)
root test (see ratio test)
limit comparison with (not sure why I thought that'd work)
Something I did try was using the fact that
to rewrite as
However, I'm not sure what to compare this to. According to wolfram alpha this series "diverges by the comparison test", but comparison to what? There is a similar problem in Baby Rudin, but for , and a simple nth root test resolves that series [convergence] in a hurry. Any ideas? Have any of you encountered such an easy looking series before? Thanks.