Reference no: EM131081622
Math 104: Homework 11-
1. Find a sequence of integrable functions (fn) on R where -∞∫∞fn = 1 for all n, but fn → 0 uniformly on R.
2. (a) By using simple properties of sin x and cos x, show how to define the function tan : (-π/2, π/2) → R. Prove that it is differentiable, strictly increasing, and neither bounded above nor below.
(b) By using inverse function theorems, define tan-1: R → (-π/2, π/2) and show that
(tan-1)'(x) = 1/1 + x2.
(c) Prove that for |x| < 1,
tan-1x = n=0∑∞((-1)nx2n+1/2n + 1.
(d) By making use of Abel's theorem, or otherwise, show that
π/4 = n=0∑∞((-1)n/2n + 1).
3. Let In = 0∫π/2sinnx dx.
(a) Prove that I0 = π/2 and I1 = 1.
(b) Use integration by parts to prove that (n + 1)In = (n + 2)In+2 for all n ≥ 0.
(c) Prove that I2m+1 ≤ I2m ≤ (1+ 1/2m)I2m+1 for all m ∈ N, and hence that I2m/I2m+1 → 1 as m → ∞.
(d) Prove that for m ∈ N,
π/2 = (2/1)(4/3)(6/5). . . (2m/2m - 1)I2m, 1 = (3/2)(5/4)(7/6). . .(2m + 1/2m)I2m+1,
and hence that
π/2 = limm→∞(2/1)(2/3)(4/3)(4/5)(6/5)(6/7). . . (2m/2m - 1)(2m/2m + 1).
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