Lebesgue integral with sum

Question

How can we swap sum under integral; $$ \sum\limits_{n=0}^\infty\int\limits_0^{\pi/2}(1-\sqrt{\sin x})^n\cos x dx $$

Answer 1

Over the given interval $1-\sqrt{\sin x}$, as well as $\cos x$, is a non-negative function. Since: $$\frac{\cos x}{\sqrt{\sin x}}\in L^1\left((0,\pi/2)\right), $$ by the dominated convergence theorem we are allowed to swap the sum and the integral, giving: $$ S = \int_{0}^{\pi/2}\frac{\cos x}{\sqrt{\sin x}}\,dx = \int_{0}^{1}\frac{dt}{\sqrt{t}}=\color{red}{2}.$$

Answer 2

All of the integrands are nonnegative, so monotone convergence does the job very simply.