This is a continuation of Exercise 80. (a) Prove that I2m+1 I2m I2m1. Hint: sin2m+1 x
Chapter 7, Problem 81(choose chapter or problem)
This is a continuation of Exercise 80. (a) Prove that I2m+1 I2m I2m1. Hint: sin2m+1 x sin2m x sin2m1 x for 0 x 2 (b) Show that I2m1 I2m+1 = 1 + 1 2m. (c) Show that 1 I2m I2m+1 1 + 1 2m. (d) Prove that lim m I2m I2m+1 = 1. (e) Finally, deduce the infinite product for 2 discovered by English mathematician John Wallis (16161703): 2 = lim m 2 1 2 3 4 3 4 5 2m 2m (2m 1)(2m + 1)
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