Formal power series

A formal power series is a mathematical expression similar to a polynomial, but it can have infinitely many terms. It takes the form \( f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \) where \( a_n \) are coefficients and \( x \) is a variable. Unlike regular functions, we don't worry about convergence for power series; we treat them as symbolic expressions. Formal power series are useful in combinatorics and algebra for encoding sequences and solving problems where you want to manipulate these series without having them tied to specific values.