I'm supposed to compute the following limit:
$$\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $$
I'm looking for a resonable approach in this case, if possible. Thanks.
I'm supposed to compute the following limit:
$$\lim_{n\to\infty} \frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\frac{\pi}{2}} (\sin x+\cos x)^n dx $$
I'm looking for a resonable approach in this case, if possible. Thanks.
Laplace's method yields $$\frac{\sqrt n}{\sqrt {2}^{n}}\int_{0}^{\pi/2} (\sin x+\cos x)^n \mathrm dx=\int_{-\pi\sqrt{n}/4}^{\pi\sqrt{n}/4}\left(\cos(t/\sqrt{n})\right)^n\mathrm dt\to\int_{-\infty}^{+\infty}\mathrm e^{-t^2/2}\mathrm dt=\sqrt{2\pi}. $$
Since $\sin x + \cos x = \sqrt{2} \sin (x+\pi/4)$ we have $$ \frac{\sqrt{n}}{ \sqrt{2}^n }\int^{\pi/2}_0 (\sin x+ \cos x)^n dx = \sqrt{n} \int^{3\pi/4}_{\pi/4} \sin^n x dx. $$
Since $\int^{\pi/4}_0\sin^n x dx \leq \frac{\pi}{4} \left( \frac{1}{\sqrt{2}} \right)^n \to 0$ our limit is the same as the limit of $$2\sqrt{n} \int^{\pi/2}_0 \sin^n x dx.$$
The integral can be evaluated by integration by parts or converting it into a Beta function, and Stirling's approximation shows the limit is $\sqrt{2\pi}.$