Let In=\int0^{\pi / 2} \sin ^n x d x. (a) Show that I{2 n+2}...

Let $I_n=\int_0^{\pi / 2} \sin ^n x d x$. (a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 k}$. (b) Use Exercise 50 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 $\lim _{n \rightarrow \infty} I_{2 \varepsilon+1} / I_{2 x}=1$. (d) Use part (c) and Exercises 49 and 50 to show that $$ \lim _{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6 {7} \cdot \cdots \cdot \frac{2 n}{2 n-1} \cdot \frac{2 n}{2 n+1}=\frac{\pi}{2} $$ This formula is usually written as an infinite product: $$ \frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdots$$ 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.

Let $I_n=\int_0^{\pi / 2} \sin ^n x d x$.
(a) Show that $I_{2 n+2} \leqslant I_{2 n+1} \leqslant I_{2 k}$.
(b) Use Exercise 50 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 $\lim _{n \rightarrow \infty} I_{2 \varepsilon+1} / I_{2 x}=1$.
(d) Use part (c) and Exercises 49 and 50 to show that $$ \lim _{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6 {7} \cdot \cdots \cdot \frac{2 n}{2 n-1} \cdot \frac{2 n}{2 n+1}=\frac{\pi}{2} $$ This formula is usually written as an infinite product: $$ \frac{\pi}{2}=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdots$$
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.

Key Concepts

Recursive Integration of Trigonometric Functions

This concept involves deriving relationships between integrals of trigonometric functions raised to different powers. Often, integration by parts is used to reduce the power of the sine function in the integral, leading to a recurrence or reduction formula that expresses a higher-power integral in terms of a lower-power one. This technique facilitates the evaluation of these integrals either in closed form or in a recursive manner.

Integral Inequalities and Bounding Techniques

This pertains to establishing upper and lower bounds for integrals, particularly those that involve functions raised to a power. By comparing integrals with similar structures but different exponents, one can deduce inequalities that help in analyzing the behavior of the sequence of integrals as the exponent varies. Such bounding techniques are fundamental in proving convergence properties and limit results.

Infinite Products and Limit Convergence

Infinite product representations arise when a sequence of multiplicative terms converges to a well?defined limit. In the context of evaluating ratios of consecutive integrals or related sequences, recognizing that the factors tend toward unity allows one to derive the convergence of the entire product. This method provides a pathway to determine limits that involve iterative or telescoping products.

Wallis Product for Pi

The Wallis product is an infinite product formulation that represents ?/2. It emerges from the analysis of integrals involving even and odd powers of sine and captures an elegant relationship between continuous functions and discrete products. The derivation typically involves establishing recursive relations and inequalities among the integrals, ultimately leading to an infinite product whose limit equals ?/2.

Geometric Construction and Limit of Aspect Ratios

This concept involves constructing sequences of geometric figures—such as rectangles built in an alternating fashion—to model and approximate limiting behavior. By analyzing how the dimensions change with each added component, one can deduce the asymptotic ratio of width to height. This approach showcases how iterative geometric procedures can converge to specific ratios, reflecting underlying analytical limits.

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