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I have following series

$$\sum_{n=3}^{\infty} \frac{(1 + 3\sin\frac{3\pi n}{4})^n}{\ln^2(n)}x^{2n}$$

where I need to find radius of convergence $R$ and investigate convergence in edge values. I have tried both methods I know, root-factor criterion and share criterion (hope I translated it correctly) but both methods lead to unsolvable limits.

Thank you for any help.

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1 Answer 1

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Hint. As regards the radius of convergence, evaluate the limit $$\limsup_{n\to +\infty} \left(\frac{|1 + 3\sin\frac{3\pi n}{4}|^n}{\ln^2(n)}\right)^{\frac{1}{2n}}.$$

P.S. Note that if $n=8k-2$ then $|1 + 3\sin\frac{3\pi n}{4}|=4$.

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  • $\begingroup$ So I need to solve series where in numerator is $4^n$? $\endgroup$
    – Bobesh
    Jun 24, 2017 at 10:31
  • $\begingroup$ @Bobesh Find the $\limsup$ along the subsequence $n=8k-1$ $\endgroup$
    – Robert Z
    Jun 24, 2017 at 10:41
  • $\begingroup$ I dont still get it. For example when k = 1, $sin(21 \pi / 4) = -1/\sqrt(2)$ $\endgroup$
    – Bobesh
    Jun 24, 2017 at 15:11
  • $\begingroup$ @Bobesh Sorry, my fault. The subsequence is $n=8k-2$ for $k\geq 1$. $\endgroup$
    – Robert Z
    Jun 24, 2017 at 16:00

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