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

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

Recall that for $|x|$, we have

$\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n$

Then

$\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}$

which is valid for $|-x^2|=|x|^2$, or more simply $|x|$.

Finally,

$f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}$