Tuesday, March 6, 2018

calculus - Finding limlimitsntoinftyfrac1n2intfracpi20xleft(fracsinnxsinxright)4,mathrmdx



limI don't know how to solve this limit problem. I'm roughly clear about using the pinch theorem, but I haven't done it yet. Could somebody please tell me? Thanks!


Answer




From
\lim_{x\to 0}x^2\left(\frac1{\sin^4 x}-\frac1{x^4}\right)=\frac23,
for x\in(0,\frac\pi 2), there is C such that
\left|\frac{x\sin^4 nx}{\sin^4 x}-\frac{\sin^4 nx}{x^3}\right|\le \frac{C\sin^4 nx}{x}\le Cn.



Thus,
\begin{align*} \lim_{n\to\infty}\frac1{n^2}\int_0^{\frac\pi2}x\left(\frac{\sin nx}{\sin x}\right)^4\, \mathrm dx&=\lim_{n\to\infty}\frac1{n^2}\int_0^{\frac\pi2}\frac{\sin^4 nx}{x^3}\, \mathrm dx\\ &= \lim_{n\to\infty}\int_0^{\frac{n\pi}2}\frac{\sin^4 x}{x^3}\, \mathrm dx\\ &= \int_0^{\infty}\frac{\sin^4 x}{x^3}\, \mathrm dx\\ \text{(IBP)}\quad &= \int_0^{\infty}\frac{2\sin^3 x\cos x}{x^2}\, \mathrm dx\\ \text{(IBP)}\quad &= \int_0^{\infty}\frac{6\sin^2 x\cos^2 x-2\sin^4 x}{x}\, \mathrm dx\\ &= \int_0^{\infty}\frac{\cos 2x-\cos 4x}{x}\, \mathrm dx\\ \text{(Frullani)}\quad &=\ln 2. \end{align*}


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