# Limit of summation as n goes to infinity

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.

Write the sum as the Riemann sum of a function: $$\lim_{n\to\infty}\sum_{k=1}^n\frac{k^{q-1}}{n^q + k^q}=\lim_{n\to\infty} \frac1n\sum_{k=1}^n\frac{(k/n)^{q-1}}{1 + (k/n)^q}=\int_0^1\frac{x^{q-1}}{1+x^q}\,dx.$$
• It is just the definition of the Riemann integral using Riemann sums:$$\int_0^1f(x)\,dx=\lim_{n\to\infty}\frac1n\sum_{k=1}^nf(x_k)$$ where $x_k\in[(k-1)/n,k/n]$. Tipical choice is $x_k=k/n$. – Julián Aguirre Jun 24 '15 at 21:58