Question

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

Answer 1

Recall that for
`|x|<1`, 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)}`