Math 104: Homework 11 (due April 29)

1. Ross exercise 34.3

2. Ross exercise 34.12

3. Ross exercise 36.6

4. Find a sequence of integrable functions (f_{n}) on where ∫_{−∞}^{∞} f_{n} = 1 for all n, but f_{n} → 0 uniformly on .

5. (a) By using simple properties of sin x and cos x, show how to define the function tan : (−\frac{π}{2}, \frac{π}{2}) → . Prove that it is differentiable, strictly increasing, and neither bounded above nor below.

(b) By using inverse function theorems, define tan^{−1} : → (−\frac{π}{2}, \frac{π}{2}) and show

that

(tan^{−1} )′(x) = \frac{1}{1 + x^{2}} .

(c) Prove that for |x| < 1,

tan^{−1} x = ∑_{n=0}^{∞} \frac{(−1)^{n}x^{2n+1}}{2n + 1} .

(d) By making use of Abel’s theorem, or otherwise, show that

\frac{π}{4} = ∑_{n=0}^{∞} \frac{(−1)^{n}}{2n + 1} .

6. Let I_{n} = ∫_{0}^{π/2} sin^{n} x dx.

(a) Prove that I_{0} = \frac{π}{2} and I_{1} = 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 + \frac{1}{2m})I_{2m+1} for all m ∈ , and hence that I_{2m}/I_{2m+1} → 1 as m → ∞.

(d) Prove that for m ∈ ,

\frac{π}{2} = \frac{2}{1} \frac{4}{3} \frac{6}{5} … \frac{2m}{2m − 1} I_{2m}, 1 = \frac{3}{2} \frac{5}{4} \frac{7}{6} … \frac{2m + 1}{2m} I_{2m+1} ,

and hence that

\frac{π}{2} = lim_{m→∞} \frac{2}{1} \frac{2}{3} \frac{4}{3} \frac{4}{5} \frac{6}{5} \frac{6}{7} … \frac{2m}{2m − 1} \frac{2m}{2m + 1} .