Question

Give a power series representation for the function f(x) x^3/(1 + 9x^2)

Answer 1

Recall that for $|x|$, we have

$\displaystyle\frac1{1-x}=\sum_{n\ge0}x^n$

Replace $x$ with $-9x^2$ and we get

$\displaystyle\frac1{1-(-9x^2)}=\sum_{n\ge0}(-9x^2)^n=\sum_{n\ge0}(-9)^nx^{2n}$

Lastly, multiply this by $x^3$, so that

$\boxed{f(x)=\displaystyle\sum_{n\ge0}(-9)^nx^{2n+3}}$