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I tried using DCT and LCT, but in both cases the comparison test fails. Below is the series: $$\sum_{n=1}^{\infty} (-1)^n \frac {\sin(n)}{n^{0.9}}$$.

Using the absolute convergence test is the first step but from thereon proving that the series is convergent/divergent is tricky. The implementation of the maclaurin series didn't work out either. I would love to know about any other method as well.

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Hint: This series is the imaginary part of $\sum\limits_{n\geqslant1}n^{-a}z^n$ for some real positive $a$ and some complex $z\ne1$ such that $|z|=1$. Does this ring a bell?

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  • $\begingroup$ So the test suggests that the series is convergent conditionally. Any other methods that come to mind... $\endgroup$ – John Oct 11 '13 at 7:55
  • $\begingroup$ @John What for? This is the One (of course one can disguise it somewhat but the core idea stays roughly the same, I believe). $\endgroup$ – Did Oct 11 '13 at 10:55
  • $\begingroup$ I like looking at different methods :). But, I guess you're right. Thanks once again for providing a quick answer :). $\endgroup$ – John Oct 11 '13 at 12:40

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