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

Question

Compute $\sum_{n=0}^{\infty}\int_{0}^{\pi/2}(1-\sqrt{\sin(x)})^{n}\cos(x)dx$. Carefully justfiy your calculations.

I want to apply Lebesgue Dominated Convergence Theorem. Can I simply commute the summation and integration? like I did below.

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

Answer 1

We do not need the full power of the dominated convergence theorem, we may just exploit the monotone convergence theorem, since both $\cos(x)$ and $(1-\sqrt{\sin x})^n$ are non-negative and bounded on $(0,\pi/2)$, and $$ \sum_{n=0}^{N}\left(1-\sqrt{\sin x}\right)^n \leq \frac{1}{\sqrt{\sin x}} $$ as well as $$ \int_{0}^{\pi/2}\frac{\cos x}{\sqrt{\sin x}}\,dx = \int_{0}^{1}\frac{dt}{\sqrt{t}} = 2.$$