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.