$\text{4.6) Prove that the sum}$

$\sum_{n=0}^{\infty}{\int_{0}^{\pi/2}{\left(1-\sqrt{\sin x}\right)^n\cos x }dx }$

$\text{converges to a finite limit, and find its value.}$

$\text{Solution: }$

$\sum_{n=0}^{\infty}{\int_{0}^{\pi/2}{\left(1-\sqrt{\sin x}\right)^n\cos x }dx }$

$= \lim_{ N \rightarrow\infty}{\sum_{n=0}^{N}{\int_{0}^{\pi/2}{\left(1-\sqrt{\sin x}\right)^n\cos x }dx }} \text{ by definition}$

$= \lim_{ N \rightarrow\infty}{\int_{0}^{\pi/2}{\cos x\sum_{n=0}^{N}{\left(1-\sqrt{\sin x}\right)^n } }dx} \text{ because the sum is finite}$

$= \int_{0}^{\pi/2}{\cos x\sum_{n=0}^{\infty}{\left(1-\sqrt{\sin x}\right)^n } dx} \text{ by the Monotone Convergence Theorem}$

$\left(1-\sqrt{\sin x}\right)^n <1 \text{ on } \left(0,\pi/2\right] \text{ so } \cos x \sum_{n=0}^{\infty}{\left(1-\sqrt{\sin x}\right)^n} = \frac{\cos x}{\sqrt{\sin x}} \text{ is finite a.e. on a bounded domain, so the integral exists}$

$\int_{0}^{\pi/2}{\frac{\cos x}{\sqrt{\sin x}} } dx = \int_{0}^{1}{\frac{1}{\sqrt{x}} } dx = 2$

-Ben Bartle