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How can i study the character of the following series? $$ \sum_{n=1}^\infty\,\,\frac{x^n(\sin1\cdot\sin2\cdot ...\cdot\sin n)^2}{(1+x\cos^21)\cdot(1+x\cos^22)\cdot...\cdot(1+x\cos^2n)},\qquad x\in\mathbb{R}.$$

i tray in this way \begin{align*} \left|\frac{x^n(\sin1\cdot\sin2\cdot...\sin n)^2}{(1+x\cos^21)\cdot(1+x\cos^22)\cdot...\cdot(1+x\cos^2n)}\right|=\left|\frac{x^n\left(\displaystyle\prod_{k=1}^{n}\sin k\right)^2}{\displaystyle\prod_{k=1}^{n}(1+x\cos^2k)}\right|=\frac{|x|^n\left(\displaystyle\prod_{k=1}^{n}\sin k\right)^2}{\left|\displaystyle\prod_{k=1}^{n}(1+x\cos^2k)\right|}; \end{align*} observing \begin{align} \prod_{k=1}^{n}\left|(1+x\cos^2k)\right| > \prod_{k=1}^{n}1=1;\qquad |x|^n\left(\displaystyle\prod_{k=1}^{n}\sin k\right)^2 <|x|^n\qquad\qquad(1) \end{align} then \begin{align*} \frac{|x|^n\left(\displaystyle\prod_{k=1}^{n}\sin k\right)^2}{\left|\displaystyle\prod_{k=1}^{n}(1+x\cos^2k)\right|}\le|x|^n\to\mbox{converge if}\,\,\,|x|<1 \end{align*} But i'm not sure abuot point $(1)$

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What do you mean by "character"? Do you mean, for what values of $x$ does it converge? Where did you come across this series? –  Gerry Myerson Apr 16 '13 at 9:12
    
yes i maen for what values of x converge! –  DeeJay Apr 16 '13 at 9:17
1  
Good. Please edit your question accordingly, so people don't have to read the comments to understand it. Also, what tests do you know for convergence? Which ones have you tried? What happened when you tried them? –  Gerry Myerson Apr 16 '13 at 12:52
    
ok ... I've tried with the test of absolute convergence, and with the comparison test, but does not work .... –  DeeJay Apr 16 '13 at 14:16
    
I edited and posted what I did –  DeeJay Apr 16 '13 at 14:26

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