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$$ \sum_{n=1}^\infty \frac{\cos n \theta}{(\sqrt{13})^{n+1}}x^n $$

Find the radius of convergence for the above series. I have learnt to use the root test and ratio test but neither of them seem to work. I have problems manipulating.

Not sure if this is useful: $\cos z = \frac{1}{2} \left(e^{iz}+e^{-iz} \right)$

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    $\begingroup$ I would use the fact that cosine is bounded. $\endgroup$
    – echoone
    Commented Aug 25, 2012 at 14:36
  • $\begingroup$ @echoone what if $\theta$ is a complex number? In this case cosine is unbounded. $\endgroup$
    – The Substitute
    Commented Mar 28, 2015 at 13:30

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Here is a hint. For any $N\in\mathbb{Z}^+$ we have that $\sup_{n>N}(\cos(n\theta))^{1/n} = 1$. Now try the root test.

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    $\begingroup$ Is this correct if $\theta$ is a complex number? Isn't cosine of a complex argument unbounded by Liouville's Theorem? $\endgroup$
    – The Substitute
    Commented Mar 28, 2015 at 13:29
  • $\begingroup$ @TheSubstitute yes. $\endgroup$
    – John Martin
    Commented Apr 2, 2015 at 2:46

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