I am trying to solve the following:
Let $q>1$ and $n \in N$. Evaluate $\lim_{n \rightarrow +\infty} \sum_{k=1} ^n \frac{k^{q-1}}{n^q + k^q}$.
I understand that I need to first get the summation into a closed form, and then take the limit of the sum. However, I am not sure how to deal with the summation. I tried to solve the summation in Mathematica (letting k = 6) first so that I knew where to go with solving the summation, but I ended up with this, which I don't now how to work with:
(1) Am I correct in proceeding by finding the closed form of the summation, then taking the limit?
(2) Can someone suggest a strategy for finding the closed form for this summation? I have never worked with an example where I was working with an unidentified real number like this.