I am having trouble finding the limit of the series:
$$\sum_{k=1}^n \frac{k^p}{n^{p+1}}$$
using Riemann integrals, where $p>1$.
Here's what I have tried so far:
I began by using the integral $$\int_1^n \frac{x^p}{n^{p+1}} dx = \frac{1}{p+1} - \frac{1}{(p+1)n^{p+1}}$$
This approaches $\frac{1}{p+1}$ as n approaches infinity.
Next, I used Riemann integrals for the tagged partition interval ${{1, ..., n+1}}$ and the partition $Y_i = \frac{i^p}{n^{p+1}}$ for $1 \leq i \leq n$ to argue that this Riemann integral also approaches $\frac{1}{p+1}$ as n approaches infinity.
However, I am unsure if my argument is correct, as I am having trouble understanding how the delta of ${1, ..., n+1}$ fits the definition. Can someone point out where I went wrong and give me a hint for the solution?
