Calculate $\lim_{n\to \infty} \frac{\frac{2}{1}+\frac{3^2}{2}+\frac{4^3}{3^2}+...+\frac{(n+1)^n}{n^{n-1}}}{n^2}$

We want to compute: $$ \color{red}{L}=\lim_{n\to +\infty}\frac{1}{n^2}\sum_{k=1}^{n}k\cdot\left(1+\frac{1}{k}\right)^k.$$ The sequence given by $a_k=\left(1+\frac{1}{k}\right)^k $ is converging to $e$, hence $$ b_k = \frac{k a_k + (k-1) a_{k-1} + \ldots + 2 a_2 + a_1}{k+(k-1)+\ldots+2+1} $$ is converging to $e$, too, by Cesàro. Since $k+(k-1)+\ldots+1 = \frac{k(k+1)}{2}$, it follows that: $$ L = \color{red}{\frac{e}{2}}.$$