Dominated convergence theorem with function $\frac{1}{x^{\frac12+\frac1n}}\left(\sin\frac{\pi}{x}\right)^n$

Question

For $n\geqslant 3$ and $x\in(0,\infty)$, we define $$f_n(x)=\frac{1}{x^{\frac12+\frac1n}}\left(\sin\frac{\pi}{x}\right)^n.$$ I have to calculate $$\lim_{n\to\infty}\int_{[1,\infty)} f_n(x)\,dx.$$

Of course, I have to use the dominated convergence theorem. I have to bound $|f_n(x)|$ by an integrable function independent of $n$.

I can think of using the inequality $|\sin u|\leq |u|$, but this gives me

$$ |f_n(x)|\leq \frac{\pi^n}{x^{\frac12+\frac1n+n}} $$

I can use that $x\geq 1$ so $x^{1/n+n}\geq 1$, but this leaves me with the $\pi^n$ term that seems to diverge.

Could someone provide any help?

Answer 1

I think you want to bound by $$ \left(\frac{\pi}{x}\right)^n x^{-1/2-1/n} $$ so $\int_\pi^\infty$ would converge since $\pi < x$, and $\int_1^\pi$ is on a compact interval and the integrand is bounded as well...