Exercise 32, § 1

Functions of a Real Variable, III, 123

Prove Wallis' Formula

\lim_{n \to \infty} \frac{1}{\sqrt{n}} \cdot \frac{2.4.6 \ldots 2n}{1.3.5 \ldots (2n-1)} = \sqrt{\pi}

by using exerc. 31 and the inequality

\sin^{n+1} x \leq \sin^n x

for

0 \leq x \leq \pi/2

Solution

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