The diagram shows the curve with equation \(y=x^{-2}\) for \(2 \leqslant x \leqslant N\) together with a set of \((N-2)\) rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
\(\sum_{r=1}^{N} \frac{1}{r^{2}}\gt \frac{3}{2}-\frac{1}{N}+\frac{1}{N^{2}}\)
(b) Use a similar method to find, in terms of \(N\), an upper bound for \(\sum_{r=1}^{N} \frac{1}{r^{2}}\).
(c) Deduce lower and upper bounds for \(\sum_{r=1}^{\infty} \frac{1}{r^{2}}\).