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