Finding power series representation How we can show a presentation of a power series and indicate its radius of convergence?
For example how we can find a power series representation of the following function?
$$f(x) = \frac{x^3}{(1 + 3x^2)^2}$$

The singularities closest to $0$ are $x=\pm i/\sqrt{3}$ so the radius of convergence (around $x=0$) is $1/\sqrt{3}$.  The series follows from Newton's expansion
$$x^3(1+3x^2)^{-2}=\sum_{k=0}^\infty {-2 \choose k} 3^k x^{2k+3} = \sum_{k=0}^\infty (k+1)(-3)^kx^{2k+3}.$$
