Series Converge or Diverge

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.

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Re: Series Converge or Diverge

There you are, I hope it helps :D

Re: Series Converge or Diverge

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.