Problem

The diagram shows the curve with equation $y = e^{1 - x}$ for $0 \le x \le 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.

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

$$\int_0^1 e^{1 - x}\,\mathrm{d}x < U_n,$$

where

$$U_n = \frac{e - 1}{n\left(1 - e^{-\frac{1}{n}}\right)}.$$

(b) Use a similar method to find, in terms of $n$, a lower bound $L_n$ for

$$\int_0^1 e^{1 - x}\,\mathrm{d}x.$$

(c) Show that

$$\lim_{n\to\infty}(U_n - L_n) = 0.$$

(d) Use the Maclaurin’s series for $e^x$ given in the list of formulae (MF19) to find the first three terms of the series expansion of $z\left(1 - e^{-\frac{1}{z}}\right)$, in ascending powers of $\frac{1}{z}$, and deduce the value of

$$\lim_{n\to\infty}(U_n).$$