Problem

The diagram shows the curve with equation $y=\dfrac{\ln x}{x^2}$ for $x\ge2$, together with a set of $(N-2)$ rectangles of unit width.

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

$$\begin{aligned} \sum_{r=1}^{N}\frac{\ln r}{r^2}<&\,\frac{2+3\ln2}{4}\\ &\,-\frac{1+\ln N}{N}. \end{aligned}$$

(b) Use a similar method to find, in terms of $N$, a lower bound for $\displaystyle\sum_{r=1}^{N}\frac{\ln r}{r^2}$.