The diagram shows the curve with equation \(y=\left(\frac{1}{2}\right)^{x}\) for \(0 \leqslant x \leqslant 1\), 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 \(\int_{0}^{1}\left(\frac{1}{2}\right)^{x} \mathrm{~d} x\gt L_{N}\), where
\(L_{N}=\frac{1}{2 N\left(2^{\frac{1}{N}}-1\right)} .\)
(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 \(U_{N}-L_{N} \leqslant 10^{-3}\).

(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 \(\ln 2\).