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I have already proved that the series above does not converge absolutely. However, I am not sure how to check whether the series converges conditionally.

I thought about using the Alternating Series Test, but I'm not sure if I can because I don't think that the sequence $\frac{1}{n^{1+\frac{1}{n}}}$ is monotonically decreasing.

If that is the case, how should I approach showing whether the series converges conditionally or diverges? Any guidance would be appreciated.

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    $\begingroup$ "I don't think that the sequence $\frac{1}{n^{1+\frac{1}{n}}}$ is monotonically decreasing." What did you try to be sure? $\endgroup$ – Did Oct 19 '15 at 10:09
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$n^{1+1/n} = \exp \left( \left(1+\frac{1}{n} \right)\ln n \right)$ and you can see that $ \left(1+\frac{1}{n} \right)\ln n$ increases when $n\ge 2$, therefore the sequence $1/n^{1+1/n}$ decreases eventually and so you can apply alternating series test. When applying series test, first few terms are not affect the convergence of the series so you can ignore finitely many terms.

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    $\begingroup$ The function $x\mapsto\left(1+\frac1x\right)\log(x)$ is actually increasing on $x>0$ hence, in the present case, there is no need to ignore finitely many terms of the series to apply the alternating series test (but the remark is definitely worthwhile in general). $\endgroup$ – Did Oct 19 '15 at 10:08

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