calculus - Convergence test from Demidovich: $sum_{n=1}^{infty} frac{n^{n + frac{1}{n}}}{(n + frac{1}{n})^n}$

I'm just learning how to test series for convergence and have encountered this series from the Demidovich's book and I can't really decide what criteria should I use. Could you please give me some hint(s)?

$$\sum_{n=1}^{\infty} \frac{n^{n + \frac{1}{n}}}{(n + \frac{1}{n})^n}$$

Answer

Hint: Since $n^{1/n} \to 1,$ you can forget about it. We're left with an $n$th term equal to

$$\frac{n^n}{(n+1/n)^n} = \left ( \frac{1}{1 + 1/n^2} \right )^n.$$