Prove that the series with terms $\frac{n^n}{ (n+1)^{n+1}}$ diverges

Question

Prove that the series with terms $\frac{n^n}{ (n+1)^{n+1}}$ diverges.

I'd like to use the comparison test, but it doesn't work out since I can either make the numerator of the new series smaller or the denominator bigger and both methods only give me another series which converges.

Answer 1

Note that

$$\frac {n^n}{(n+1)^{n+1}}=\frac {1}{(n+1)}\frac {1}{\left(1+\frac 1n\right)^n}\begin{cases}\ge\frac {1}{e(n+1)}\\\sim \frac {1}{e(n+1)}\end{cases}$$

then use comparison test or limit comparison test with $\frac {1}{e(n+1)}$.

Answer 2

$${{n^n}\over{(n+1)^{n+1}}}={n^n\over n^{n+1}}{1\over{(1+{1\over n})^{n+1}}}$$

$$={1\over n}{1\over{(1+{1\over n})^{n+1}}}$$ compare with ${1\over n}$