A formal Laurent series over a ring 
is a series with only finitely many negative powers,
 |
where 
and 
.
The collection 
is the localization ![R[[z]][z^(-1)]](/images/equations/FormalLaurentSeries/Inline5.svg)
of the formal
power series ring. Products are defined coefficientwise by finite
sums and do not require analytic convergence. For a field 
, every nonzero series is invertible by factoring out its first
nonzero monomial and inverting the remaining power series.
Vector-valued series use the same lower-bound convention. In a vertex
operator algebra, the field applied to any fixed state has this truncation, even
though the operator coefficients themselves need not have a common lower
bound.
