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

$$1+\cos\theta+\cos2\theta+\cdots+\cos(n-1)\theta=\frac{1}{2}\left(1-\cos n\theta+\frac{\sin n\theta\sin\theta}{1-\cos\theta}\right).$$

The diagram shows the curve with equation $y=\cos x$ for $0<x\le1$, 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

$$\int_0^1\cos x\,\mathrm{d}x<\frac{1}{2n}\left(1-\cos1+\frac{\sin\frac{1}{n}\sin1}{1-\cos\frac{1}{n}}\right).$$

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

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