Q:

Sum of a (finite) hyperharmonic series

For some $k < \infty$, and $p \in (0,1)$, consider the following sum:
$S_k = \sum_{n=1}^{k} \frac{1}{n^p}$. 
What is a closed form expression for $S_k$? 

A:

I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral:
$$\sum_{n=1}^kn^{-p}\approx1+\int_1^k x^{-p}dx=1+\left.\frac1{1-p}x^{1-p}\right\rvert^k_0=1+\frac{k^{1-p}}{1-p}.$$

