Editing Laurent series (section)

== Multiplication and sum ==
Laurent series cannot in general be multiplied.
Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the [[convolution]] of integer sequences).
Geometrically, the two Laurent series may have non-overlapping annuli of convergence.

Two Laurent series with only ''finitely'' many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at <math>c</math>, and inner radius of convergence 0, so they both converge on an overlapping annulus.

Thus when defining [[Formal power series#Formal Laurent series|formal Laurent series]], one requires Laurent series with only finitely many negative terms.

Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.

Also, for a field <math>F</math>, by the sum and multiplication defined above, [[formal Laurent series]] would form a field <math>F((x))</math> which is also the field of fractions of the ring <math>F[[x]]</math> of [[formal power series]].