Unit of power series ring

Question

Is there any way to calculate the multiplicative group of the units of power series ring $k[[x]]$ where $k$ is a field

Answer 1

Hint $\rm\displaystyle\quad 1\: =\: (a-xf)(b-xg)\ \Rightarrow\ ab=1$ $$\Rightarrow\ \ \displaystyle\rm\frac{1}{a-xf}\ =\ \frac{b}{1-bxf}\ =\ b\:(1+bxf+(bxf)^2+(bxf)^3+\:\cdots\:)$$

Answer 2

$\bf Hint:$ $\sum_{n=0}^\infty a_nx^n$ is a unit iff $a_0\ne 0$.

Answer 3

The multiplicative group is $k[[x]]\backslash (x)$. Certainly those elements divisible by $x$ are not units. If an element is not divisble by $x$ (in other words, has nonzero constant term), you can construct the inverse term by term.