Limit $\frac{\frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+...+\frac{(n+1)^n}{n^{n-1}}}{n^2}$ [duplicate]
After applying Stolz-Cesàro we get
$$ \lim_{n\to\infty}\frac{\sum_{k=1}^n\frac{(k+1)^k}{k^{k-1}}}{n^2} = \lim_{n\to\infty}\frac{\frac{(n+1)^n}{n^{n-1}}}{n^2-(n-1)^2} = \lim_{n\to\infty}\frac{(\frac{n+1}{n})^{n}}{2-1/n} = e/2 $$