The diagram shows the curve with equation \(y=\frac{\ln x}{x^{2}}\) for \(x \geqslant 2\), together with a set of \((N-2)\) rectangles of unit width.
(a) By considering the sum of the areas of these rectangles, show that
\(\sum_{r=1}^{N} \frac{\ln r}{r^{2}}\lt \frac{2+3 \ln 2}{4}-\frac{1+\ln N}{N} .\)
(b) Use a similar method to find, in terms of \(N\), a lower bound for \(\sum_{r=1}^{N} \frac{\ln r}{r^{2}}\).