Let In y y2 0 sinn x dx. (a) Show that I2n12 < I2n11 < I2n. (b) Use Exercise 50 to show

Chapter 7, Problem 74

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Let In y y2 0 sinn x dx. (a) Show that I2n12 < I2n11 < I2n. (b) Use Exercise 50 to show that I2n12 I2n 2n 1 1 2n 1 2 (c) Use parts (a) and (b) to show that 2n 1 1 2n 1 2 < I2n11 I2n < 1 and deduce that limnl` I2n11yI2n 1. (d) Use part (c) and Exercises 49 and 50 to show that lim nl` 2 1 ? 2 3 ? 4 3 ? 4 5 ? 6 5 ? 6 7 ? ? 2n 2n 2 1 ? 2n 2n 1 1 2 This formula is usually written as an infinite product: 2 2 1 ? 2 3 ? 4 3 ? 4 5 ? 6 5 ? 6 7 ? 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.