Editing Formal power series (section)

=== Multiplicative inverse ===
The series

:<math>A = \sum_{n=0}^\infty a_n X^n \in R[[X]]</math>

is invertible in <math>R[[X]]</math> if and only if its constant coefficient <math>a_0</math> is invertible in <math>R</math>. This condition is necessary, for the following reason: if we suppose that <math>A</math> has an inverse <math>B = b_0 + b_1 x + \cdots</math> then the [[constant term]] <math>a_0b_0</math> of <math>A \cdot B</math> is the constant term of the identity series, i.e. it is 1. This condition is also sufficient; we may compute the coefficients of the inverse series <math>B</math> via the explicit recursive formula

:<math>\begin{align}
b_0 &= \frac{1}{a_0},\\
b_n &= -\frac{1}{a_0} \sum_{i=1}^n a_i b_{n-i}, \ \ \ n \geq 1.
\end{align}</math>

An important special case is that the [[geometric series]] formula is valid in <math>R[[X]]</math>:

:<math>(1 - X)^{-1} = \sum_{n=0}^\infty X^n.</math>

If <math>R=K</math> is a field, then a series is invertible if and only if the constant term is non-zero, i.e. if and only if the series is not divisible by <math>X</math>. This means that <math>K[[X]]</math> is a [[discrete valuation ring]] with uniformizing parameter <math>X</math>.