Formal power series

A formal power series is an infinite sequence of coefficients written as powers of a variable and manipulated algebraically without requiring convergence.

Cauchy product: The multiplication rule for series, in which each coefficient is a finite sum of pairwise coefficient products. It defines multiplication using only finitely many terms for each resulting coefficient.
Commutative ring: A set with addition and multiplication satisfying the ring laws, including commutative multiplication. Formal series inherit their coefficient arithmetic from a commutative ring.
Generating function: A power series whose coefficients encode a sequence, often to turn sequence operations into algebra. Formal series let counting identities work without convergence assumptions.
Convergent power series: A power series that converges for at least one nonzero value of its variable. Formal series need not converge at any value; their coefficients alone define them.
Composition of formal power series: Substitution of one formal power series into another, defined when the substituted series has zero constant term. Zero constant term ensures each output coefficient depends on only finitely many input terms.
Polynomial ring: A ring of finite sums of powers of variables with coefficients in a given ring. Polynomials are the finite expressions that formal series extend.
Ordinary generating function: The series Σₙ aₙxⁿ used to encode a sequence without factorial weights. It is the standard formal-series encoding for many counting sequences.
Analytic function: A function locally represented by a convergent power series. Analytic functions give numerical meaning to series where formal series remain algebraic.
Formal derivative: The coefficient operation sending each term aₙxⁿ to naₙxⁿ⁻¹. Termwise differentiation is defined algebraically and requires no convergence.
Truncated polynomial ring: A polynomial ring in which powers at or above a fixed degree are set to zero. Reducing series modulo powers of the variable gives finite polynomial approximations.

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