How to find the limit of this seq.?

Question

Please help me to solve this limit:

$$ \lim _{n\to \infty }\left(\frac{1}{n^2}\sqrt[n^2]{e}+\frac{2}{n^2}\sqrt[n^2]{e^4}+\frac{3}{n^2}\sqrt[n^2]{e^9}+...+\frac{n}{n^2}\sqrt[n^2]{e^{n^2}}\right) $$
Thank you

Answer 1

You just need to write it precise with $\Sigma$, then it's easier to see.

The sequence is equal to:

$\displaystyle\sum_{k=1}^n\frac{k}{n^2}\sqrt[n^2]{e^{k^2}} = \sum_{k=1}^n \frac{1}{n}\frac{k}{n}e^{\frac{k^2}{n^2}} \longrightarrow \int_0^1x e^{x^2}dx=\color{red}{\frac{e-1}{2}}\qquad (n\to\infty)$