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Does $ \displaystyle \int_{0}^{\infty} \ \frac{\sin (\tan x)}{x} dx $ converge?

$ \displaystyle \int_{0}^{\infty} \frac{\sin (\tan x)}{x} \ dx = \int_{0}^{\frac{\pi}{2}} \frac{\sin (\tan x)}{x} \ dx + \sum_{n=1}^{\infty} \int_{\pi(n-\frac{1}{2})}^{\pi(n+\frac{1}{2})} \frac{\sin (\tan x)}{x} \ dx $

The first integral converges since $\displaystyle \frac{\sin (\tan x)}{x}$ has a removable singularity at $x=0$ and is bounded near $ \displaystyle \frac{\pi}{2}$.

And $ \displaystyle \int_{\pi(n-\frac{1}{2})}^{\pi(n+\frac{1}{2})} \frac{\sin (\tan x)}{x} \ dx$ converges since $\displaystyle \frac{\sin (\tan x)}{x}$ is bounded near $\pi(n-\frac{1}{2})$ and $\pi(n+\frac{1}{2})$.

But does $ \displaystyle \sum_{n=1}^{\infty} \int_{\pi(n-\frac{1}{2})}^{\pi(n+\frac{1}{2})} \frac{\sin (\tan x)}{x} \ dx$ converge?

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Yes.

Note that $$\begin{align} I_n:=\int_{\pi(n-\frac12)}^{\pi(n+\frac12)}\frac{\sin(\tan(x))}{x}\,\mathrm dx&=\int_0^{\frac\pi2}\left(\frac1{n\pi+x}-\frac1{n\pi-x}\right)\sin(\tan(x))\,\mathrm dx\\&=\int_0^{\frac\pi2}\frac{-2x}{n^2\pi^2-x^2}\sin(\tan(x))\,\mathrm dx\\ \end{align}$$ With $A:=\int_0^{\frac\pi2}\max\{-2x\sin(\tan (x)),0\}\,\mathrm dx$, $B:=\int_0^{\frac\pi2}\min\{-2x\sin(\tan (x)),0\}\,\mathrm dx$ (which both converge), we can thus estimate $$ \frac{1}{n^2\pi^2-\pi^2/4}B+\frac{1}{n^2\pi^2-0}A\le I_n\le \frac{1}{n^2\pi^2-\pi^2/4}A+\frac{1}{n^2\pi^2-0}B,$$ The difference between these bounds and $\frac1{n^2\pi^2}(A+B)$ is governed by $$\frac{1}{n^2\pi^2-\frac{\pi^2}{4}}-\frac1{n^2\pi^2}=\frac1{4\pi^2 n^4-\pi^2 n^2},$$ hence $$ I_n=\frac1{n^2\pi^2}(A+B)+O(n^{-4}).$$ We conclude that $\sum I_n$ converges at least as good as $\sum\frac1{n^2}$.

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  • $\begingroup$ Oops, there is a sign error - back to the drawing board $\endgroup$ – Hagen von Eitzen May 29 '13 at 21:01
  • $\begingroup$ Don't worry. I'm not in a hurry. $\endgroup$ – user26647 May 29 '13 at 21:03
  • $\begingroup$ @FractionalDerivative With corrected sign we get convergence like $\sum\frac1{n^2}$ instead of divergence like $\sum\frac1n$. I think that makes you happier :) $\endgroup$ – Hagen von Eitzen May 29 '13 at 21:09
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    $\begingroup$ Even we can say more: $$ \int_{0}^{\infty} \frac{\sin (\tan x)}{x} \, dx = \frac{\pi}{2} \log(1 - e^{-1}). $$ Refer to here for a proof. $\endgroup$ – Sangchul Lee May 30 '13 at 0:41

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