Problem

The diagram shows the curve with equation $y=x^{-2}$ for $2\le x\le N$ together with a set of $(N-2)$ rectangles of unit width.

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

$$\sum_{r=1}^{N}\frac{1}{r^2}>\frac{3}{2}-\frac{1}{N}+\frac{1}{N^2}.$$

(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

$$\sum_{r=1}^{\infty}\frac{1}{r^2}.$$