$\sum_{n=1}^{+\infty}(3^{1/n^2} -1) \csc^2n$
It's obviously a positive series and the parts of $\csc^2n $ tells me it might be divergent. But I have no idea how to prove it. Any one can offer some help or hints?
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Please see this question. – Paramanand Singh Mar 25 '21 at 03:01
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Thank you! I've been searching for quite a while. I'll take a quick look – Shirai Yanagi Mar 25 '21 at 03:17
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The only answer in the other question is pretty much what I thought, The potential divergence of the series comes from the occasional but irrational spikes provided by $csc^2(n)$, but it's still of little help for a rigorous proof. – Shirai Yanagi Mar 25 '21 at 03:41
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You can see that the problem essentially needs information on irrationality measure of $\pi$ and I am not sure if the problem has been solved or not. – Paramanand Singh Mar 25 '21 at 03:42
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hmm nope, I still have no idea how to prove the series is divergent.. – Shirai Yanagi Mar 25 '21 at 12:58