The diagram shows the curve with equation \(y=\frac{1}{x^{2}+1}\) for \(0 \leqslant x \leqslant 1\), 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 \(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{n}{n^{2}+r^{2}}\).