How does one show $\sum_{n \leq x/M} \frac{1}{n^{\delta}} \approx \frac{(N/M)^{\delta}}{1-\delta}$, where $n \leq N$ and $\delta>0$.
I assume this has something to do with Taylor series, but im not sure around what number.
It also might be useful to use the Euler-Mac Laurin summation formula...