Problem

The diagram shows the curve with equation $y = \frac{1}{x^2 + 1}$ for $0 \le x \le 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.

(a) By considering the sum of the areas of these rectangles, show that

$$\sum_{r = 1}^{n} \frac{n}{n^2 + r^2} < \frac{1}{4}\pi.$$

(b) Use a similar method to find a lower bound for $\sum_{r = 1}^{n} \frac{n}{n^2 + r^2}$. Give your answer in terms of $n$ and $\pi$.

(c) Deduce the exact value of $\lim_{n \to \infty}\sum_{r = 1}^{n} \frac{n}{n^2 + r^2}$.