How to calculate a infinite Riemann sum $\lim\limits_{n\to \infty} \sum\limits_{i=1}^n \frac{n}{i^2+n^2}$

Question

I am working on this assignment and I got a little stuck up with this. I got some ideas but I not at all sure if they are correct. So I am hoping to get opinion from someone in here please.

How to calculate this in detail?

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{n}{i^2+n^2}$$

Answer 1

Hint: The key is to identify the sum as a Riemann sum for a certain definite integral. Then you can do the integration rather than trying to compute the sum. Try rewriting the summand in terms of $i/n$.

Answer 2

$$\lim_{n \rightarrow \infty} \sum_{i=1}^n \frac{n}{i^2+n^2} = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \frac{1}{1+\frac{i^2}{n^2}}$$

This is in the form of a Riemann sum, which, in this limit, becomes

$$\int_0^1 dx \: \frac{1}{1+x^2} $$

You should recognize the integrand as the derivative of $\arctan{x}$, and you can take it from here.