Let $I_n=\int_0^{\pi / 2} \sin ^n x d x$.
(a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 n}$.
(b) Use Exercise 56 to show that
$$
\frac{I_{2 n+2}}{I_{2 n}}=\frac{2 n+1}{2 n+2}
$$
(c) Use parts (a) and (b) to show that
$$
\frac{2 n+1}{2 n+2} \leqslant \frac{I_{2 n+1}}{I_{2 n}} \leqslant 1
$$
and deduce that $\lim _{n \rightarrow \infty} I_{2 n+1} / I_{2 n}=1$.
(d) Use part (c) and Exercises 55 and 56 to show that
$$
\lim _{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots \cdot \frac{2 n}{2 n-1} \cdot \frac{2 n}{2 n+1}=\frac{\pi}{2}
$$
This formula is usually written as an infinite product:
$$
\frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots
$$
and is called the Wallis product.
(e) We construct rectangles as follows. Start with a square of area 1 and attach rectangles of area 1 alternately beside or on top of the previous rectangle (see the figure). Find the limit of the ratios of width to height of these rectangles.
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