Question

Find a power series representation for the function. (Give your power series representation centered at x = 0.)f(x) = x3x2 + 1f(x) = ∞n = 0 Determine the interval of convergence. (Enter your answer using interval notation.)

Answer 1

I suppose you mean

`f(x)=(x^3)/(x^2+1)`

Recall that for
`|x|<1`, we have

`\frac1{1-x}=\displaystyle\sum_(n=0)^\infty x^n`

Then

`\frac1{1+x^2}=\frac1{1-(-x^2)}=\displaystyle\sum_(n=0)^\infty(-x^2)^n=\sum_(n=0)^\infty(-1)^nx^(2n)`

which is valid for
`|-x^2|=|x|^2<1`, or more simply
`|x|<1`.

Finally,

`f(x)=\displaystyle(x^3)/(x^2+1)=\sum_(n=0)^\infty(-1)^nx^(2n+3)`