The Wallis Product Formula for π Let(a) Show that I2n+2 ≤ I2n+1

The Wallis Product Formula for π Let

(a) Show that I_2n+2 ≤ I_2n+1 ≤ I_2n.

(b) Use Exercise 56 to show that

(c) Use parts (a) and (b) to show that

and deduce that

(d) Use part (c) and Exercises 55 and 56 to show that

This formula is usually written as an infinite product:

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.

Data From Exercise 55:

(a) Use the reduction formula to show that

where n ≥ 2 is an integer.

(b) Use part (a) to evaluate ∫^π/2₀ sin³x dx and ∫^π/2₀ sin⁵x dx.

(c) Use part (a) to show that, for odd powers of sine,

Data From Exercise 56:

Prove that, for even powers of sine,

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