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.