Using an Already Obtained Riemann Sum to answer another question

Question

I have an upper Riemann Sum of a question which I have gotten to be $\sum_{i=1}^{n}\frac{n}{n^{2}+i^{2}}$ on the interval [0,1]. switching the $i$ to a summation I transformed this to $\lim_{n \to \infty} \frac{6}{2n^{2}+9n+1}.$ Using this result i need to show $\lim_{n \to \infty} \sum_{i=0}^{n-1} \frac{n}{n^{2}+j^{2}} = \frac{\pi}{4}$. I have no idea how i can show this any help? The original function is $F(x) = \frac{1}{1+x^{2}}$.

Answer 1

$$\lim_{n\to +\infty}\sum_{j=0}^{n-1}\frac{n}{n^2+j^2} = \lim_{n\to +\infty}\frac{1}{n}\sum_{j=0}^{n-1}\frac{1}{1+\left(\frac{j}{n}\right)^2}=\int_{0}^{1}\frac{dx}{1+x^2}=\arctan(1)=\frac{\pi}{4}.$$