Problem

The diagram shows part of the curve $y=x\operatorname{sech}^2x$ and its maximum point $M$.

(a) Show that, at $M$,

$$2x\tanh x-1=0$$

and verify that this equation has a root between $0.7$ and $0.8$.

(b) By considering a suitable set of rectangles, use the diagram to show that

$$\sum_{r=2}^{n}r\operatorname{sech}^2r<n\tanh n+\ln(\operatorname{sech}n)-\tanh 1-\ln(\operatorname{sech}1).$$