Convergence of sequences of periodic functions

Question

Given $f_n(x) = F(x) (\cos(\pi x))^n$, where $n \in \mathbb N , F: \mathbb R \to \mathbb R$ integrable.

Simple question: does the Lebesgue integral converge in $\mathbb R$? (and how to show?)

Answer 1

Does $\cos^n x$ converge? (Yes). Find its limit and apply the dominated convergence theorem.