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I would like to understand how to improve the quality of my question:Doubts on asymptotic criterion for $\sum_{n=1}^{\infty}a_n=\sum_{n=1}^{\infty}n^{a}\tan^{-1}\bigg(\frac{1}{n^a}\bigg)-e^{1/n}$ with $a>0$.
Indeed I have received only some comments (without any conclusion), even if I have shown my attempt and I have underlined my doubts and the troubling passages.
Why do you think my question is without an answer? Thank you in advance!

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    $\begingroup$ looks ok to me. I think either less people who know what you're asking fluently or general disinterest from dealing with such Qs $\endgroup$ May 19 at 10:26

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