My attempt:
I did Root Test and got '1' the test failed, and I couldn't do the Logarithmic Test.
My textbook has done it in solved examples but its wrong. Book says convergent but it is divergent(wolfram alpha).
$$\frac{n^n}{(n+1)^{n+1}}=\frac{1}{(1+\frac{1}{n})^n}\times\frac{1}{n+1}$$ Note that $(1+\frac{1}{n})^n<e$ for any positive integer $n$ so $\frac{1}{(1+\frac{1}{n})^n}>\frac{1}{e}$. Therefore $$\sum_{n=1}^{\infty}\frac{n^n}{(n+1)^{n+1}}=\sum_{n=1}^{\infty}\frac{1}{(1+\frac{1}{n})^n}\times\frac{1}{n+1}>\frac{1}{e}\sum_{n=1}^{\infty}\frac{1}{n+1}$$ where the last sum diverges.
