There you are, I hope it helps
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.
Thank you for this!
I found the answer through another recently, though. One way to show divergence was to use the inequality for all and . Then substituting gives
Therefore the series diverges. Your solution is very nice because it gives a direct verification of the necessary inequality for the comparison test, which I find a bit more intuitive than "find the right Analysis inequality to get the job done" kind of argument.