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I have the series $$\sum_{n=0}^\infty (-1)^n 2n x^{(2n -1)}$$
It turns out that this series is equal to the function $$\frac{-2x}{(1+x^2)^2}$$

Is there a general method that would demonstrate this fact beforehand? I'm looking for an algorithm that expresses a power series as a rational function whenever the power series converges to a rational function.

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There is no single method for summing all power series. However, there is the sum of a geometric series: $$ \sum_{k=0}^\infty x^k=\frac1{1-x} $$ Substitution yields $$ \sum_{k=0}^\infty(-1)^kx^{2k}=\frac1{1+x^2} $$ Taking the derivative gives $$ \sum_{k=0}^\infty(-1)^k2kx^{2k-1}=\frac{-2x}{(1+x^2)^2} $$

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