TITLE: How do I evaluate $\lim_{n\to\infty} \,\sum_{k=1}^n\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}$?
QUESTION [7 upvotes]: I came across the following problem recently in a problem sheet aimed at high school students:

Evaluate $$\lim_{n\to\infty} \,\sum_{k=1}^n\left(\frac{k}{n^2}\right)^{\frac{k}{n^2}+1}.$$

I tried to rewrite the inner sum as a Riemann sum hoping that the limit would become a definite integral, but no gain because of the extra $1/n$'s.

REPLY [2 votes]: Adding to @NinadMunshi 's answer
For those who didn't understand why, for $n>e$, $$\left(\frac{1}{n}\right)^{\frac{1}{n}} < \left(\frac{k}{n^2}\right)^{\frac{k}{n^2}} < \left(\frac{1}{n^2}\right)^{\frac{1}{n^2}}$$, the graph of $f(x)=x^x$ starts increasing after $x=1/e$ for $f^\prime(x)=(1+\ln x)e^{x\ln x}=(1+\ln x)x^x$.