Hello,

the problem is to calculate inverse element (reciprocal) of $\displaystyle 1-x^3$ in ring $\displaystyle $\mathbb{R}[[x]]$.

I skimmed some Wikipedia and found following formula:

Inverting series

But I don't know to use it. Also the following model answer didn't help me either:

"Task is to calculate inverse element (reciprocal) of $\displaystyle 1-x$ in the ring $\displaystyle $\mathbb{R}[[x]]$.

So, we have to calculate $\displaystyle A(x)=a_0+a_1x+a_2x^2+...,$ that

$\displaystyle (1-x)(a_0+a_1x+a_2x^2+...)=1$.

When the $\displaystyle a_0$ is solved, and then $\displaystyle a_1$..., we see that inverse element is $\displaystyle 1+x+x^2+...$"

I don't understand how those $\displaystyle a_0$ etc. are solved. Only thing, that is clear to me is, that $\displaystyle 1+x+x^2+...$ is generated by $\displaystyle \frac{1}{1-x}$, which equals $\displaystyle (1-x)^{-1}$.

So, any help is appreciated. Thanks very much!