The diagram shows the curve with equation \(y=\frac{1}{\sqrt{x}} \mathrm{e}^{\sqrt{x}}\) 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
\(\sum_{r=1}^{n} \frac{1}{\sqrt{r}} \mathrm{e}^{\sqrt{r}}<\left(2+\frac{1}{\sqrt{n}}\right) \mathrm{e}^{\sqrt{n}}-2 \mathrm{e} .\)
(b) Use a similar method to find, in terms of \(n\), a lower bound for \(\sum_{r=1}^{n} \frac{1}{\sqrt{r}} \mathrm{e}^{\sqrt{r}}\).