TITLE: Convergence of $\sum_{n=1}^{\infty}n^{-{(1+1/n)}}$
QUESTION [5 upvotes]: I'm having difficulties to determine if the series 
$$\sum_{n=1}^{\infty}n^{-{(1+1/n)}}$$
converges or not. 
I have rewritten the series in the form 
$$\sum_{n=1}^{\infty}\frac{1}{n^{(1+1/n)}}=\sum_{n=1}^{\infty}\frac{1}{n\sqrt [n]{n}}$$
but I don't know how I should continue form here.

REPLY [2 votes]: Yet another method. Let $a_n = \frac{1}{n \sqrt[n]{n}}$. Note that $a_n$ is positive and decreasing. Therefore, by Cauchy's Condensation Theorem, one has 
$$\sum_{n=1}^{\infty}\frac{1}{n \sqrt[n]{n}} \sim \sum_{n=1}^{\infty}2^n\frac{1}{2^n (2^n)^{\frac{1}{2n}}}=\sum_{n=1}^{\infty}\frac{1}{\sqrt{2}}$$
which clearly diverges. 
If you are not familiar with this test look here: Cauchy Condensation Test.