Formal power series

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In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.).

A formal power series is a special kind of formal series, of the form

${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots ,}$

where the ${\displaystyle a_{n},}$ called coefficients, are numbers or, more generally, elements of some ring, and the ${\displaystyle x^{n}}$ are formal powers of the symbol ${\displaystyle x}$ that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.

A formal power series with coefficients in a ring ${\displaystyle R}$ is called a formal power series over ${\displaystyle R.}$ The formal power series over a ring ${\displaystyle R}$ form a ring, commonly denoted by ${\displaystyle R[[x]].}$ (It can be seen as the (x)-adic completion of the polynomial ring ${\displaystyle R[x],}$ in the same way as the p-adic integers are the p-adic completion of the ring of the integers.)

Formal powers series in several indeterminates are defined similarly by replacing the powers of a single indeterminate by monomials in several indeterminates.

Formal power series are widely used in combinatorics for representing sequences of integers as generating functions. In this context, a recurrence relation between the elements of a sequence may often be interpreted as a differential equation that the generating function satisfies. This allows using methods of complex analysis for combinatorial problems (see analytic combinatorics).