Evaluate: $$\lim_{n \to \infty} \frac{\sqrt[n]{e} + 2\sqrt[n]{e^2} + \cdots + n\sqrt[n]{e^n}}{n^2}$$
I can see a pattern in the numerator, but I don't know how to solve this limit.
Evaluate $\lim_{n \to \infty} \frac{\sqrt[n]{e} + 2\sqrt[n]{e^2} + \cdots + n\sqrt[n]{e^n}}{n^2}$
3
$\begingroup$
calculus
limits
2 Answers
7
Hint we can write it as
$$ \lim_ {n \to \infty} \sum _{r=1}^n \frac {r\cdot e^{r/n}}{n^2} $$
now thats limit as a sum convert it to integral with $r/n=x ,1/n=dx $ thus it converts to $$\int _0 ^1 x.e^xdx $$ thus the limit is evaluation of this integral by parts .
4
This is a Riemann limit: $$\int_0^1te^tdt=\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac kne^{k/n}$$