TITLE: Convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)^{1+\frac{1}{n}}$
QUESTION [6 upvotes]: I have problem in determining the convergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)^{1+\frac{1}{n}}$. It seems like it is convergent given that $(1+\frac{1}{n})>1$ for all n, but I still cannot prove it rigorously. 
Can anyone help me ??

REPLY [8 votes]: Note that
$$ \lim_{n\to\infty} n^{1/n} = 1. $$
Thus we have
$$ \lim_{n\to\infty} \frac{\left(\dfrac{1}{n^{1+1/n}}\right)}{\left(\dfrac{1}{n}\right)}= 1. $$
Now you can apply the limit comparison test to conclude that the series diverge to $+\infty$.