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Obviously for $t=0$ the integral diverges.

For $t>1$ I have tried to subsitute $u=x^t$ and then split the integral up into segments of pi length such that I can write it as an alternating sum. Then (I think) I managed to prove that I can use Leibniz test since the alternating integrals get ever so smaller as $x \to \infty$.

Also for $x\in(0,1]$ I can bound the sine from below by inscribing ever increasing triangles thus proving that the sequence doesn't go to 0, which implies that the series is divergent.

Now for $t<0$ I am completely clueless, partly because the graph is vastly different in that case. Substituting $1/x = u$ gets ugly very quickly, so I didn't follow through.

I'd highly appreciate any ideas and solutions, particularly if they are more elegant than my tries so far have been :)

Thanks in advance!

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  • $\begingroup$ You might want to look at this and this. $\endgroup$ – Mattos Nov 3 '18 at 15:24
  • $\begingroup$ Unfortunately I must not use the Dirichlet for integrals for I have not learned it yet. But thank you for the great resources - odd that I didn't find them myself. $\endgroup$ – whiterock Nov 3 '18 at 15:28
  • $\begingroup$ Ignore the second link, the first one gives you what you want. There is also a good way to search the site for specific math text, I believe it was written by a user of the site who got annoyed with the in built MSE search function. It's called approach0. $\endgroup$ – Mattos Nov 3 '18 at 15:29
  • $\begingroup$ As I said, alas I cannot use the Dirichlet test. $\endgroup$ – whiterock Nov 3 '18 at 15:33
  • $\begingroup$ The first link doesn't use the Dirichlet test? $\endgroup$ – Mattos Nov 3 '18 at 15:37

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