TITLE: How to calculate $ \lim_{n\to\infty} (2^n+3^n+\cdots+n^n)^{1/n}/n ?$
QUESTION [5 upvotes]: I need help in calculating the following limit.

$$\lim_{n\to\infty}\frac{\sqrt[n]{2^n+3^n+\cdots +n^n}}{n}$$

REPLY [7 votes]: $$\frac{\sqrt[n]{2^n+3^n+\cdots +n^n}}{n}=\frac{n\sqrt[n]{\left(\frac{2}{n}\right)^n+\left(\frac{3}{n}\right)^n+\cdots +1}}{n}=\sqrt[n]{\left(\frac{2}{n}\right)^n+\left(\frac{3}{n}\right)^n+\cdots +1}$$
$$1<\sqrt[n]{\left(\frac{2}{n}\right)^n+\left(\frac{3}{n}\right)^n+\cdots +1}<\sqrt[n]{n}\underset{n\to{\infty}}\to{1}$$