Calculating $ \lim_{n\rightarrow \infty} \sum_{k=1}^n \frac{k}{n^2}e^{\frac{k^2}{n^2}}$

Question

I'm trying to calculate the following limit:

$$ \lim_{n\rightarrow \infty} \sum_{k=1}^n \frac{k}{n^2}e^{\frac{k^2}{n^2}}$$

I tried getting to a form so I can use Riemann's sum but I can see which partition I should use.

Any suggestions?

Answer 1

Hint: This is $$ \frac1n\sum \frac k n e^{(\frac k n)^2}. $$ Can you see what function has this as a Riemann sum? ($x=\frac k n$)