# Finding the limit of a series. [duplicate]

I haven't touched series and sequences in a while and am rusty, so need some direction in finding the following:

$$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}$$

• Write $\frac{k}{n^2+k^2}=\frac{1}{n}\frac{k/n}{1+(k/n)^2}$ and see that your sum is a Riemann sum on $[0,1]$ for the uniform partition. – user569197 Jul 3 '18 at 11:24
Use Riemann sum: $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}=\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{\frac{k}{n}}{1 + \left(\frac{k}{n}\right)^2}\frac{1}{n}=\int_0^1\frac{x}{1+x^2}dx$$