I'm having some trouble solving the following limit:
$$\lim_{n \to +\infty} \frac{\sqrt[n] e + \sqrt[n] {e^2} + ... + \sqrt[n] {e^n}}{n}$$
This question is in the "Riemann Sum" section so I think that we're supposed to turn this into an integral, so:
$$\lim_{n \to +\infty} \frac{\sqrt[n] e + \sqrt[n] {e^2} + ... + \sqrt[n] {e^n}}{n} = \lim_{n \to +\infty} \sum_{k=1}^n \dfrac{1}{n} \sqrt [n] {e^k} = \int_a^b f(x) dx$$
I think that $n$ is the number of partitions and $1/n$ is the length of each one, so this means that $b - a = 1$ or $b = a+1$, meaning that we only need to find a value for $a$ and $b$ will be that $+1$. But now I can't seem to find the value of $a$ nor $f(x)$. How can I solve this?
Note that $\sqrt[n]{e^k}=e^{k/n}$ and that therefore$$\lim_{n\to\infty}\frac1n\sum_{k=1}^ne^{k/n}=\int_0^1e^x\,\mathrm dx=e-1.$$