Proving $\lim_{n \rightarrow \infty} \frac{\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$ How do we prove that $$\lim_{n \rightarrow \infty} \dfrac{\displaystyle\sum_{r=1}^{n} r^a}{n^{a+1}}=\frac{1}{a+1}$$?
This type of terms appear in problems on limits, but I am unable to prove this.
Please help me out.

As $$\lim_{n \to \infty} \frac1n\sum_{r=1}^n f\left(\frac rn\right)=\int_0^1f(x)dx,$$
we have $$\lim_{n\to\infty}\frac1n\sum_{r=1}^n\left(\frac rn\right)^a=\int_0^1x^a\ dx$$
