Editing Formal power series (section)

==Introduction==
A formal power series can be loosely thought of as an object that is like a [[polynomial]], but with infinitely many terms. Alternatively, for those familiar with [[power series]] (or [[Taylor series]]), one may think of a formal power series as a power series in which we ignore questions of [[Convergent series|convergence]] by not assuming that the variable ''X'' denotes any numerical value (not even an unknown value). For example, consider the series
<math display=block>
A = 1 - 3X + 5X^2 - 7X^3 + 9X^4 - 11X^5 + \cdots.
</math>
If we studied this as a power series, its properties would include, for example, that its [[radius of convergence]] is 1 by the [[Cauchy–Hadamard theorem]]. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of [[coefficient]]s [1, −3, 5, −7, 9, −11, ...]. In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the [[factorial]]s [1, 1, 2, 6, 24, 120, 720, 5040, ... ] as coefficients, even though the corresponding power series diverges for any nonzero value of ''X''.

Algebra on formal power series is carried out by simply pretending that the series are polynomials. For example, if

:<math>B = 2X + 4X^3 + 6X^5 + \cdots,</math>

then we add ''A'' and ''B'' term by term:

:<math>A + B = 1 - X + 5X^2 - 3X^3 + 9X^4 - 5X^5 + \cdots.</math>

We can multiply formal power series, again just by treating them as polynomials (see in particular [[Cauchy product]]):

:<math>AB = 2X - 6X^2 + 14X^3 - 26X^4 + 44X^5 + \cdots.</math>

Notice that each coefficient in the product ''AB'' only depends on a ''finite'' number of coefficients of ''A'' and ''B''. For example, the ''X''<sup>5</sup> term is given by

:<math>44X^5 = (1\times 6X^5) + (5X^2 \times 4X^3) + (9X^4 \times 2X).</math>

For this reason, one may multiply formal power series without worrying about the usual questions of [[absolute convergence|absolute]], [[conditional convergence|conditional]] and [[uniform convergence]] which arise in dealing with power series in the setting of [[Mathematical analysis|analysis]].

Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series ''A'' is a formal power series ''C'' such that ''AC'' = 1, provided that such a formal power series exists. It turns out that if ''A'' has a multiplicative inverse, it is unique, and we denote it by ''A''<sup>−1</sup>. Now we can define division of formal power series by defining ''B''/''A'' to be the product ''BA''<sup>−1</sup>, provided that the inverse of ''A'' exists. For example, one can use the definition of multiplication above to verify the familiar formula

:<math>\frac{1}{1 + X} = 1 - X + X^2 - X^3 + X^4 - X^5 + \cdots.</math>

An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator <math>[X^n]</math> applied to a formal power series <math>A</math> in one variable extracts the coefficient of the <math>n</math>th power of the variable, so that <math>[X^2]A=5</math> and <math>[X^5]A=-11</math>. Other examples include

:<math>\begin{align}
\left[X^3\right] (B) &= 4, \\ 
\left[X^2 \right] (X + 3 X^2 Y^3 + 10 Y^6) &= 3Y^3, \\
\left[X^2Y^3 \right] ( X + 3 X^2 Y^3 + 10 Y^6) &= 3, \\
\left[X^n \right] \left(\frac{1}{1+X} \right) &= (-1)^n, \\ 
\left[X^n \right] \left(\frac{X}{(1-X)^2} \right) &= n.
\end{align}</math>

Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.