Problem

(a) State the sum of the series $1+z+z^2+\cdots+z^{n-1}$, for $z\ne1$.

(b) By letting $z=\cos\theta+\mathrm{i}\sin\theta$, where $\cos\theta\ne1$, show that

\begin{aligned} 1+\cos\theta+\cos 2\theta+\cdots+\cos(n-1)\theta =&\,\frac{1}{2}\left(1-\cos n\theta\\ &\,+\frac{\sin n\theta\sin\theta}{1-\cos\theta}\right). \end{aligned}

The diagram shows the curve with equation $y=\cos x$ for $0\le x\le1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.

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

$$\begin{aligned} \int_0^1\cos x\,\mathrm{d}x<&\,\frac{1}{2n}\Bigg(1-\cos 1\\ &\,+\frac{\sin 1\sin\left(\frac{1}{n}\right)}{1-\cos\left(\frac{1}{n}\right)}\Bigg). \end{aligned}$$

(d) Use a similar method to find, in terms of $n$, a lower bound for $\int_0^1\cos x\,\mathrm{d}x$.