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I am trying to prove that $$\lim_{n\rightarrow \infty}\int_{a}^{\pi/2}\sqrt{n} \cos^n(x) dx=0\hspace{10mm}\forall a\in \left(0,\frac{\pi}{2}\right)$$.

My trial

I am thinking of using Dominated or monotone convergence theorem. Since the limit goes to $0$ from non zero value, $\{\sqrt{n}\cos^n\}$ should be somehow decreasing. Thus, the way seems to lean toward dominated convergence theorem.

But even if I can somehow put the limit inside of integral, I am not sure if $$\lim_{n\rightarrow \infty} \sqrt{n}\cos^n(x)=0 \hspace{4mm} a.e.$$ is true.

I hope I can get any hint from this site :) Thank you in advance!

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You can just see that $\cos$ is decreasing from $a$ to $\frac{\pi}{2}$. So you have :

$$ \int_a^{\pi/2} \sqrt{n}\cos(x)^n dx \leqslant \sqrt{n}\cos(a)^n (\pi/2 - a) \underset{n\rightarrow\infty}{\longrightarrow 0} $$

since $\cos(a) \in [0,1)$ and by compared growths.

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  • $\begingroup$ Thank you so much! $\endgroup$ – Lev Ban Aug 21 '18 at 14:16

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