Can you help me with the convergence of the series
$$\sum_{n=1}^{\infty} \left( \frac{n}{n+1} \right)^{n(n+1)}$$
I tried the ratio limit but i couldn't obtain the limit, do you have an idea?, the limit of the ratio test is
$$\lim_{n \rightarrow \infty} \left(\frac{n+1}{n} \right)^{n(n+1)} \left(\frac{n+1}{n+2}\right)^{(n+1)(n+2)}$$
I review it on Wolframalpha, and it's suppose to converge to $\frac{1}{e}$ but I don't understand why.
Best Answer
What about the root test?
\begin{align*} \lim_{n\to\infty}(a_{n})^{1/n} & = \lim_{n\to\infty}\left(\frac{n}{n+1}\right)^{n+1}\\\\ & = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{-n} = e^{-1} < 1 \end{align*}
Thus the proposed series converges, and we are done.
Hopefully this helps!