The Art of Proof pp 161-165 | Cite as

# Generating Functions

## Abstract

Assume \((a_n )_{n = 0}^\infty\) is a sequence of real numbers that is not explicitly defined by a formula, for example, a recursive sequence. One can sometimes get useful information about this sequence—identities, formulas, etc.— by embedding it in a **generating function**: \(A(x): = \sum\limits_{n - 0}^\infty {a_n x^n}.\) We use the convention that the members of the sequence are named by a lowercase letter and the corresponding generating function is named by its uppercase equivalent. We think of this series *A(x)* as a *formal power series*, in the sense that questions of convergence are ignored. So operations on formal power series have to be defined from scratch.

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