One of my friends gave me the following question from his review, I have little experience to dealing with these types of questions in Analysis so if you could help us just to get started it would be quite helpful.
Rigorously justify the following: $\int_{0}^{1} \frac{dx}{1+x^2} = \lim_{N \to \infty} \sum_{n=0}^{N} \frac{(-1)^n}{2n+1}$