Does $\int_1^\infty\sin (\frac{\sin x}{x})\mathrm d x$diverge or not? If it converges, does it converge conditionally or absolutely? I guess that it converges conditionally, also,I think it may be related to $\int_{n\pi}^{(n+1)\pi}\frac{\sin x}{x}\mathrm d x$ , but I do not know how to start? Any help will be appreciated.
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1$\begingroup$ One hint is that sin(x) is bounded below and above by -1 and 1. What will this tell us about sin(x)/x? $\endgroup$ – mathreadler Jun 14 '15 at 16:16
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1$\begingroup$ @mathreadler: In itself that doesn't tell us enough -- for example, $\cos(1/x)$ has the same bounds, but $\cos(1/x)/x$ does not have a bounded integral. $\endgroup$ – hmakholm left over Monica Jun 14 '15 at 16:18
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$\begingroup$ Hint: For $x$ large, the function behaves as $\sin(x)/x$.. $\endgroup$ – tired Jun 14 '15 at 16:34
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$\begingroup$ yes it is not enough in itself. it is just intended to be a hint. $\endgroup$ – mathreadler Jun 14 '15 at 17:27
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For any $n\in\mathbb{Z}^+$ the integral $\int_{0}^{+\infty}\left(\frac{\sin x}{x}\right)^n\,dx$ can be explicitly computed (see here, for instance).
From the approximation $\frac{\sin x}{x}\approx e^{-x^2/6}$, it is expected to be positive and decay like $\sqrt{\frac{3\pi}{2n}}$.
These facts already, together with $\sin z$ being an entire function, give that our integral is a converging one. A more convincing argument, maybe, is that for any $x\geq 1$ the inequality: $$ \left(1-\frac{1}{5x^2}\right)\cdot\frac{\sin x}{x}\leq \sin\left(\frac{\sin x}{x}\right)\leq\frac{\sin x}{x} $$ holds. Being equivalent to $\int_{0}^{+\infty}\frac{\sin x}{x}\,dx$, our integral is converging but not absolutely converging.
answered Jun 14 '15 at 17:20
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$\begingroup$ What does your inequality imply? I am confused about it, could your please explain it in detail? $\endgroup$ – Lorence Jun 15 '15 at 6:51
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$\begingroup$ @Lorence: a function between two Riemann-integrable functions is Riemann-integrable. Then take absolute values and check that the integral is not converging absolutely. $\endgroup$ – Jack D'Aurizio Jun 15 '15 at 10:50
