Problem

The diagram shows the curve with equation $y = \left(\frac{1}{2}\right)^x$ for $0 \le x \le 1$, together with a set of $N$ rectangles each of width $\frac{1}{N}$.

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

$$\int_0^1 \left(\frac{1}{2}\right)^x\,\mathrm{d}x > L_N,$$

where

$$L_N = \frac{1}{2N\left(2^{\frac{1}{N}} - 1\right)}.$$

(b) Use a similar method to find, in terms of $N$, an upper bound $U_N$ for

$$\int_0^1 \left(\frac{1}{2}\right)^x\,\mathrm{d}x.$$

(c) Find the least value of $N$ such that $U_N - L_N \le 10^{-3}$.

(d) Given that

$$\int_0^1 \left(\frac{1}{2}\right)^x\,\mathrm{d}x = \frac{1}{2\ln 2},$$

use the value of $N$ found in part (c) to find upper and lower bounds for $\ln 2$.