How to find limit of sum $\lim\limits_{n\to\infty}\sum_{k=1}^{100n}\frac{k^p}{n^{p+1}}$ How can I find this limit? I've tried to use Stolz theorem, but have not succeed. I have heard smth about Riemann sums, but have not found good algorithm how to use it. Can you help me to solve it with the help of riemann sums or show me algorithm how to use it


I thought it might be instructive to present an approach that does not use Riemann sums.  To that end, we proceed.


Note that
$$\sum_{k=1}^N \underbrace{\left(k^{p+1}-(k-1)^{p+1}\right)}_{=(p+1)k^p+O(k^{p-1})}=N^{p+1}$$
which by induction reveals that 
$$\sum_{k=1}^N k^p=\frac{N^{p+1}}{p+1}+O(N^p)\tag1$$
Hence, we have
$$\sum_{k=1}^{100n}\frac{k^{p}}{n^{p+1}}=\frac{100^{p+1}}{p+1}+O\left(\frac1n\right)$$
Now let $n\to\infty$
