I am trying to evaluate this limit but I am unsuccessful: $$ \lim_{n \to \infty} \frac{1+\sqrt[n]{e}+\sqrt[n]{e^2}+\cdots++\sqrt[n]{e^{n-1}}}{n}$$ I have to do it with Riemann sum (by converting it to a definite integral), but I am stuck. I wrote it like that (I hope that's correct): $$ \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \sqrt[n]{e^{k-1}} $$
I guess I should take the intervals to have length $\frac{1}{n}$, but I don't know how to find the function $f(x)$ to integrate. What bothers me the most is that $n$-th root that I don't know what to do with. I was also thinking of somehow applying $\ln$, but I don't know how.
The sum is of the form \begin{align*} \dfrac{1}{n}\sum_{k=1}^{n}e^{\frac{k-1}{n}}. \end{align*} It is of the left endpoint taken with the partition $\{0,1/n,...,(n-1)/n,1\}$, so it converges to the integral $\int_{0}^{1}e^{x}dx$.