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


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

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.