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Possible Duplicate:
Convergence/divergence of $\int_0^{\infty}\frac{x-\sin x}{x^{7/2}}\ dx$

Determine the improper integral $\int_0^{\infty} \frac{x-\sin x}{x^{7/2}}dx$ converge or diverge. Prove that please.

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marked as duplicate by rschwieb, user26872, draks ..., Micah, Austin Mohr Dec 12 '12 at 19:54

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  • $\begingroup$ "infinity" should be "\infty" and do you mean $x^{7/2}$ or $\frac{x^7}{2}$? (if its the first, enclose $7/2$ in curly braces: x^{7/2} $\endgroup$ – icurays1 Dec 12 '12 at 18:30
  • $\begingroup$ You need to analyze the badness near $0$. My approach would be the Taylor series for sine. Informally, near $0$ the top behaves like $x^3/6$. $\endgroup$ – André Nicolas Dec 12 '12 at 18:31
  • $\begingroup$ Why did you have to ask the same question twice? $\endgroup$ – Nameless Dec 12 '12 at 18:31
  • $\begingroup$ @icurays1 x^(7/2) $\endgroup$ – user315 Dec 12 '12 at 18:32
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Write for $0< \varepsilon<1$ $$\int\limits_{0}^{+\infty}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}=\int\limits_{0}^{\varepsilon}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}+\int\limits_{\varepsilon}^{1}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}+\int\limits_{1}^{+\infty}{\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\, dx}.$$ Since for small $x, \;\; 0<x<\varepsilon$ $$x-\sin{x}=\dfrac{x^3}{3!}+O(x^5), $$ then $$\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\sim {\dfrac{1}{3!}}x^{3-\frac{7}{2}}={\dfrac{1}{3!}}x^{-\frac{1}{2}},$$ therefore the first integral in RHS converges.

The second integral is proper integral, therefore it is finite.

The third integral converges, since $\left|\dfrac{x-\sin{x}}{x^{\frac{7}{2}}}\right| \leqslant \dfrac{x+1}{x^{\frac{7}{2}}} \leqslant \dfrac{2x}{x^{\frac{7}{2}}}=2x^{-\frac{5}{2}}.$

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  • $\begingroup$ I think you mean to write $x - \sin x = x^3/3! + O(x^5)$, since it does not make sense to say that something is $o(\cdots)$ on a fixed interval. $\endgroup$ – Antonio Vargas Dec 12 '12 at 19:48
  • $\begingroup$ @Antonio Vargas Thanks! Edited. $\endgroup$ – M. Strochyk Dec 12 '12 at 19:52

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