Problem

The diagram shows the curve with equation $y = \frac{1}{\sqrt{x}}e^{\sqrt{x}}$ for $x \ge 1$, together with a set of $n - 1$ rectangles of unit width.

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

$$\sum_{r = 1}^{n} \frac{1}{\sqrt{r}}e^{\sqrt{r}} < \bigg(2 + \frac{1}{\sqrt{n}}\bigg)e^{\sqrt{n}} - 2e.$$

(b) Use a similar method to find, in terms of $n$, a lower bound for $\sum_{r = 1}^{n} \frac{1}{\sqrt{r}}e^{\sqrt{r}}$.