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?
$$f(x) = \frac{x^3}{(1 + 3x^2)^2}$$

1) Write down the long familiar power series representation of $\dfrac{1}{1-t}$.
2) Differentiate term by term to get the power series representation of $\dfrac{1}{(1-t)^2}$.
3) Substitute $-3x^2$ everywhere that you see $t$ in the result of 2). 
4) Multiply term by term by $x^3$.
For the radius of convergence, once you have obtained the series, the Ratio Test will do the job. Informally, our orginal geometric series converges when $|t|\lt 1$. So the steps we took are OK if $3x^2\lt 1$, that is, if $|x|\lt \frac{1}{\sqrt{3}}$. 
