TITLE: Approximation of a summation by an integral
QUESTION [11 upvotes]: I am going to approximate $\sum_{i=0}^{n-1}(\frac{n}{n-i})^{\frac{1}{\beta -1}}$ by $\int_{0}^{n-1}(\frac{n}{n-x})^{\frac{1}{\beta -1}}dx$, such that $n$ is sufficiently large.

Is the above approximation true?
If the above approximation is true, by which theorem or method (like Newton's method) it can be holds? What is the error of approximation?

REPLY [2 votes]: Using Riemann Siegel formula you can approximate $\sum_{i=0}^{n-1} \left( \frac{n}{n-i} \right)^s =\simeq n^s\left\{\zeta(s)-1/(s-1)(n+1/2)^{1-s}\right\}.$
This gives very accurate values for real $s>1$ even for very small $n$. For instance, for $n=2$ and $s=3$, one obtains $\zeta(3)\simeq 241/200= 1.205$.