evaluating sum of limits

Question

I was given this problem in definite integral and limits chapter $$\lim_{n \to \infty}\left( \frac{n}{n^2 + 1^2}+ \frac{n}{n^2 + 2^2} + \frac{n}{n^2 + 3^2 }+........+\frac{n}{n^2 + n^2}\right)$$ which method is used to evaluate these kind of problems and how do i evaluate this sum of limits ? I have heard some Riemann's sum is used but i do not know it and how to solve by that method if you can explain me how its done

Answer 1

Hint: You can Use Riemann sums.

$$ \sum_{k=1}^{n}\frac{n}{n^2+k^2}= \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+(k/n)^2}\longrightarrow_{n\to \infty} \int_{0}^{1}\frac{1}{1+x^2}dx=\dots.$$