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How to prove that :

$$ \sum_{n=1}^{\infty}\frac{1}{2nx^{2n}}=-\frac{1}{2}\ln \left(1-\frac{1}{x^2}\right)$$

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    Do you know that $\sum \frac1nx^n = -\ln(1-x)$ for $|x|<1$?2012-09-20
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    Other useful facts for this problem: $1 - 1/x^2 = (1 - 1/x)(1 + 1/x)$ and $\ln(ab) = \ln(a) + \ln(b)$.2012-09-20

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