Find a sequence of integrable functions fn on r where

Question

Math 104: Homework 11-

1. Find a sequence of integrable functions (f_n) on R where _-∞∫^∞f_n = 1 for all n, but f_n → 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 + x)².

(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.

(e) Optional for the enthusiasts. Calculate (5 + i)⁴(239 - i) and use it to prove Machin's formula

π/4 = 4 tan-1 (1/5) - tan-1(1/239).

3. Let I_n = ₀∫^π/2 sinⁿx dx.

(a) Prove that I₀ = π/2 and I₁ = 1.

(b) Use integration by parts to prove that (n + 1)I_n = (n + 2)I_n+2 for all n ≥ 0.

(c) Prove that I_2m+1 ≤ I_2m ≤ (1+ 1/2m))I_2m+1 for all m ∈ N, and hence that I_2m/I_2m+1 → 1 as m → ∞.

(d) Prove that for m ∈ N,

π/2 = (2/1)(4/3)(6/5). . . (2m/2m - 1)I_2m,               1 = (3/2)(5/4)(7/2). . .(2m + 1/2m)I2m+1,

and hence that

π/2  = lim_m→∞(2/1)(2/3)(4/3)(4/5)(6/5)(6/7). . . (2m/2m - 1)(2m/2m + 1).