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Limit with Integration

jacks

Well-known member
Apr 5, 2012
226
Finding $$\lim_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(z)dz$$
 

jacks

Well-known member
Apr 5, 2012
226
Solution
Put
\begin{equation*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}\cos^{2n-2}(x)\,\mathrm{d}x.
\end{equation*}
Via the substitutions $ y=\sin x $ and $ y=\frac{z}{\sqrt{n-1}} $ we get
\begin{gather*}
I_{n}=\sqrt{n}\int_{0}^{\pi/4}(1-\sin^2(x))^{n-1}\,\mathrm{d}x = \sqrt{n}\int_{0}^{1/\sqrt{2}}(1-y^2)^{n-1}\cdot\dfrac{1}{\sqrt{1-y^2}}\,\mathrm{d}y =\\[2ex]
\dfrac{\sqrt{n}}{\sqrt{n-1}}\int_{0}^{\sqrt{n-1}\left/\sqrt{2}\right.}\left(1-\dfrac{z^2}{n-1}\right)^{n-1}\cdot\dfrac{1}{\sqrt{1-\dfrac{z^2}{n-1}}}\,\mathrm{d}z = \dfrac{\sqrt{n}}{\sqrt{n-1}}\int_{0}^{\infty}f_{n(z)}\,\mathrm{d}z
\end{gather*}
where
\begin{equation*}
f_{n}(z)=\begin{cases}
\left(1-\dfrac{z^2}{n-1}\right)^{n-1}\cdot\dfrac{1}{\sqrt{1-\dfrac{z^2}{n-1}}}&\mbox{ if } 0<z<\sqrt{n-1}\left/\sqrt{2}\right.\\
0&\mbox{ if } z>\sqrt{n-1}\left/\sqrt{2}\right.
\end{cases}
\end{equation*}


Then $ 0 \le f_{n}(z)<e^{-z^2}\cdot \dfrac{1}{\sqrt{1-1/2}} $ and $\displaystyle \lim_{n\to \infty}f_{n}(z) = e^{-z^2}.$


Consequently, according to Lebesgue's dominated convergence theorem
\begin{equation*}
\lim_{n\to \infty}I_{n} = \int_{0}^{\infty}e^{-z^2}\,\mathrm{d}z =\dfrac{\sqrt{\pi}}{2}.
\end{equation*}


**Remark.** This is an alternative answer where we use the beta function and the gamma function. From


https://en.wikipedia.org/wiki/Beta_function


we get
\begin{equation*}
\sqrt{n}\int_{0}^{\pi/2}\cos^{2n-2}(x)\,\mathrm{d}x = \dfrac{\sqrt{n}\,\Gamma(n-\frac{1}{2})}{\Gamma(n)}\cdot\dfrac{\sqrt{\pi}}{2}\to \dfrac{\sqrt{\pi}}{2}, \mbox{ as } n\to \infty
\end{equation*}
where we find the limit here https://en.wikipedia.org/wiki/Gamma_function


Since
\begin{equation*}
0 \le \sqrt{n}\int_{\pi/4}^{\pi/2}\cos^{2n-2}(x)\,\mathrm{d}x \le \sqrt{n}\,2^{1-n}\cdot\dfrac{\pi}{4} \to 0, \mbox{ as } n\to \infty
\end{equation*}