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

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up vote 3 down vote accepted

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 $|x|\lt \frac{1}{\sqrt{3}}$.

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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

$$x^3(1+3x^2)^{-2}=\sum_{k=0}^\infty {-2 \choose k} 3^k x^{2k+3} = \sum_{k=0}^\infty (k+1)(-3)^kx^{2k+3}.$$

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