How to calculate $ \lim_{n\to\infty} (2^n+3^n+\cdots+n^n)^{1/n}/n ?$

Question

I need help in calculating the following limit.

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

Answer 1

HINT

Note that $n^n < 2^n + 3^n + \cdots + n^n < n \cdot n^n$. Now use squeeze theorem to get the limit.

Answer 2

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