Dummit and Foote 7.2.3(c) - units of R[[x]] the formal power series I'm having trouble solving the part C of the problem on Dummit and Foote 7.2.3
Prove that 
$$\alpha = \sum_{n=0}^\infty  a_nx^n$$
is a unit in $R[x]$ if and only if $a_0$ is a unit in $R$.
I had no trouble proving the direction where we assume $\sum_{n=0}^\infty  = a_nx^n$ is a unit in $R$, but I have no clue about how to go about the other direction.
My approach was to construct an element $\beta$ in $R[[x]]$ such that $\alpha\beta = 1$, but not sure how to proceed to do so.
Any help would be greatly appreciated!

For $f(x) \in R[[x]]$ be a unit in $R[[x]]$, it must exists $g(x) \in R[[x]]$ such that
$$f(x)g(x)=(\sum_{n=0}^{\infty}a_nx^n)(\sum_{n=0}^{\infty}b_nx^n)=\sum_{n=0}^{\infty}(\sum_{k=0}^{n}a_kb_{n-k})x^n=1.$$
We see that this is possible in and only if $a_0b_0=1$, $a_0b_1+a_1b_0=0$, ..., $\sum_{k=0}^{n}a_{k}b_{n-k}=0$, for all  $n\in\mathbb{N}$. So, this is possible if and only if $a_0$ is a unit in $R$, with inverse $b_0$. In that caseWe solve $a_0b_0=1$, then we solve $a_0b_1+a_1b_0=0$, and go on...
