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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1One hint is that sin(x) is bounded below and above by -1 and 1. What will this tell us about sin(x)/x? – mathreadler Jun 14 '15 at 16:16
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1@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. – hmakholm left over Monica Jun 14 '15 at 16:18
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Hint: For $x$ large, the function behaves as $\sin(x)/x$.. – tired Jun 14 '15 at 16:34
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yes it is not enough in itself. it is just intended to be a hint. – 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.
Jack D'Aurizio
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What does your inequality imply? I am confused about it, could your please explain it in detail? – Faye Tao Jun 15 '15 at 06:51
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@Lorence: a function between two Riemann-integrable functions is Riemann-integrable. Then take absolute values and check that the integral is not converging absolutely. – Jack D'Aurizio Jun 15 '15 at 10:50
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We have $\sin x\leq x\iff x\geq0$, so your inequality $$\sin\left(\frac{\sin x}{x}\right)\leq\frac{\sin x}{x},\quad x>1$$ seems not right! – Riemann Apr 16 '24 at 04:26
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Use the following limit:
$$\lim_{x\to0}\frac{x-\sin x}{x^3}=\frac16.$$
So we have
$$\lim_{x\to\infty}\frac{\frac{\sin x}{x}-\sin\left(\frac{\sin x}{x}\right)}{\frac{\sin^3x}{x^3}}=\frac16,$$
this implies
$$\int_{1}^{\infty}\left[\frac{\sin x}{x}-\sin\left(\frac{\sin x}{x}\right)\right]{\rm d} x$$
is absolutely convergent(because $\int_{1}^{\infty}\frac{\sin^3x}{x^3}{\rm d}x$ is absolutely convergent).
Also
$$\int_{1}^{\infty}\frac{\sin x}{x}{\rm d}x$$
is conditional convergent.
So we get
$$\int_{1}^{\infty}\sin\left(\frac{\sin x}{x}\right){\rm d} x$$
is conditional convergent.
Riemann
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