Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$
converge?
Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series
$\sum\limits_{k=1}^{\infty}\Big(\frac{k}{k+1}\Big)^k$ diverges to $+\infty$. I find hard to determine if $n$ or the sum goes faster to $+\infty.$
Thanks in advance.
Best Answer
Recall that by Stolz-Cesaro we have
$$\lim_{n\to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=L \implies \lim_{n\to \infty} \frac{a_n}{b_n}=L$$
and in that case we have
$$\lim_{n\to \infty}\frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}=\lim_{n\to \infty}\frac{1}{\Big(\frac{n+1}{n+2}\Big)^{n+1}}=\lim_{n\to \infty}{\Big(\frac{n+2}{n+1}\Big)^{n+1}}=\lim_{n\to \infty}{\Big(1+\frac{1}{n+1}\Big)^{n+1}}\to e$$
To determine the rate of convergence and also as an alternative to solve the limit, we have that
$${\Big(\frac{k}{k+1}\Big)^k}=e^{k\log \Big(\frac{k}{k+1}\Big)}=e^{-k\log \Big(1+\frac{1}{k}\Big)}=e^{-k\Big(\frac{1}{k}-\frac{1}{2k^2}+O(k^{-3})\Big)}e^{-1+\frac{1}{2k}+O(k^{-2})}=\frac1e\left(1+\frac{1}{2k}+O(k^{-2})\right)$$
and therefore
$$\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k=\frac1e\sum\limits_{k=1}^{n}\left(1+\frac{1}{2k}+O(k^{-2})\right)\sim\frac1e\left(n+\frac12\ln n\right)$$