Troubles with a limit sequence.

Question

I need to calculate the following limit $$ L=\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{n}{n^2+i^2} $$

I get that $\frac{1}{2}\leq L\leq1$ and I think that $L=\frac{1}{2}$ but I can't prove it.

Can someone give me a hint?

Answer 1

$$\sum_{k=1}^\infty\frac n{n^2+k^2}=\frac1n\sum_{k=1}^\infty\frac1{1+\left(\frac kn\right)^2}\xrightarrow[n\to\infty]{}\int_0^1\frac{dx}{1+x^2}=\ldots$$

The result is not $\;\cfrac12\;$ .

Answer 2

Hint

You have: $$\sum_{i=1}^n \frac{n}{n^2+i^2}=\frac{1}{n} \sum_{i=1}^n \frac{1}{1+\left(\frac{i}{n} \right)^2}$$ You can then notice that the sum look like a Riemann sum and: $$\int_0^1 \frac{dx}{1+x^2}=\arctan(1)=\frac{\pi}{4}$$