Sunday, 22 January 2017

real analysis - Compute limntoinftyintfracpi20sumnk=1(sinx)kdx




Compute lim





I tried writing \displaystyle \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^n (\sin{x})^kdx = \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{1-(\sin{x})^{n+1}}{1-\sin{x}}dx, but I don't know how to continue form here.



I know that the limit diverges to \infty, but I don't know how to prove it.


Answer



The wanted limit is +\infty. Indeed, by letting I_k=\int_{0}^{\pi/2}\left(\sin x\right)^k\,dx we have that \{I_k\}_{k\geq 1} is a decreasing sequence, but it is also log-convex by the Cauchy-Schwarz inequality, and by integration by parts
I_{k}^2\geq I_k I_{k+1} = \frac{\pi}{2(k+1)}\tag{1}
such that
\sum_{k=1}^{n}I_k \geq \sqrt{\frac{\pi}{2}}\sum_{k=1}^{n}\frac{1}{\sqrt{k+1}}\geq \sqrt{\frac{\pi}{2}}\sum_{k=1}^{n}2\left(\sqrt{k+2}-\sqrt{k+1}\right)=\sqrt{2\pi}\left(\sqrt{n+2}-\sqrt{2}\right).\tag{2}


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