Evaluating the following sum

Question

I have no idea how to solve evaluate this integral:

$$\lim_{n\to\infty} \frac{1^a + 2^a + \cdots + n^a}{n^{1+a}}, a > -1$$

I want to set this up as some sort of integration since it is a large sum but I am not sure how to go about this. Any help would be appreciated!

Answer 1

$$\lim_{n\to\infty}\frac{\displaystyle\sum_{k=1}^nk^a}{n^{a+1}}=\lim_{n\to\infty}\frac1n\cdot\sum_{k=1}^n\bigg(\frac kn\bigg)^a=\int_0^1x^adx=\bigg[\frac{x^{a+1}}{a+1}\bigg]_0^1=\frac1{a+1}$$