Editing Series (mathematics) (section)

=== Formal power series ===
{{main|Formal power series}}

While many uses of power series refer to their sums, it is also possible to treat power series as ''formal sums'', meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in [[combinatorics]] to describe and study [[sequence]]s that are otherwise difficult to handle, for example, using the method of [[generating function]]s.  The [[Hilbert–Poincaré series]] is a formal power series used to study [[graded algebra]]s.

Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as [[addition]], [[multiplication]], [[derivative]], [[antiderivative]] for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a [[commutative ring]], so that the formal power series can be added term-by-term and multiplied via the [[Cauchy product]]. In this case the algebra of formal power series is the [[total algebra]] of the [[monoid]] of [[natural numbers]] over the underlying term ring.<ref>{{citation|author=Nicolas Bourbaki|author-link=Nicolas Bourbaki|title=Algebra|publisher=Springer|year=1989}}: §III.2.11.</ref>  If the underlying term ring is a [[differential algebra]], then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.