Does $\sum _n \int_0^{\frac{\pi }{2}} \cos ^n(x) \, dx$ converge? I've tried appling: https://en.wikipedia.org/wiki/Monotone_convergence_theorem
and ended up with:
$\int_0^{\frac{\pi }{2}} \frac{1}{1-\cos (x)} \, dx$ and an assumption that $\left| \cos (x)\right| <1$
did I mess sth up, or this sum simply does not converge?
(tried using Mathematica, but found nothing worth mentioning)

Since all the involved functions are non-negative, there is indeed equivalence between the convergence of $\sum _n \int_0^{\frac{\pi }{2}} \cos ^n(x) \, dx$ and that of $\int_0^{\frac{\pi }{2}} \frac{1}{1-\cos (x)} \, dx$. For the latter, the potential problems is at zero. We can decide, by looking at an equivalent of $\frac{1}{1-\cos (x)}$ near zero (which follows from the definition of derivative).
