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Here's the question:

$$\sum_{n=0}^{\infty}\frac{x^{\frac{3}{2}}}{\left ( 1+x^{2} \right )^{n}}= \left\{\begin{matrix} 0 &\left (x=0 \right ) \\ x^{\frac{-1}{2}}+x^{\frac{3}{2}} & \left ( 0< x\leq 1 \right ) \end{matrix}\right.$$

Show that this is true. (I'd be glad if the approach is constructive, instead of backtracking by Taylor series). And also, can we find a general formula for

$$\sum_{n=0}^{\infty}\frac{x^\alpha }{\left ( 1+x^{\beta } \right )^{n}}$$

Thanks.

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    Oh, I didn't realize that this was a simple power series. Sorry about the question.2012-03-07
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    Simple *geometric* series may be what you mean.2012-03-07
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    @Gerry Myerson yes :)2012-03-07
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    @firemind $$\sum\limits_{n \geqslant 0} {\frac{{{x^a}}}{{{{\left( {1 + {x^b}} \right)}^n}}}} = {x^{a - b}} + {x^a}$$2012-03-07

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