The Wallis Product Formula for Let (a) Show that I 2n+2 I 2n+1 I

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The Wallis Product Formula for π Let

(a) Show that I2n+2 ≤ I2n+1 ≤ I2n.

(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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Related Book For  book-img-for-question

Calculus Early Transcendentals

ISBN: 9781337613927

9th Edition

Authors: James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

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