This is a continuation of Exercise 76. Data From Exercise 76 Set Im = /2 0

This is a continuation of Exercise 76.

Data From Exercise 76

Set Im = ∫^π/2₀ sin^m x dx. Use Exercise 69 to prove that

Conclude that

Data From Exercise 69

Let I_m = ∫^π/2₀ sin^m x dx.

Show that I₀ = π/2 and I₁ = 1.
Prove that, for m ≥ 2,

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(a) Prove that I2m+1 ≤ 12m ≤ 12m-1. sin2m+1 12m-1 12m+1 (c) Show that 1 ≤ (b) Show that = 1 + (d) Prove that lim m-∞0 1 2m ≤1+ 12m 12m+1 12m 12m+1 (e) Finally, deduce the infinite product for discovered by English mathematician John Wallis (1616-1703): = 1. 1 2m KIN x ≤ sin2m π x≤ sin2m-1. X for 0≤x≤ 1/ 2244 3 3 5 = lim 2 m-00 1 2m. 2m (2m 1)(2m + 1)