Editing Formal power series (section)

=== Formal differentiation ===
Given a formal power series

:<math>f = \sum_{n\geq 0} a_n X^n \in R[[X]],</math>

we define its '''[[formal derivative]]''', denoted ''Df'' or ''f'' ′, by

:<math> Df = f' = \sum_{n \geq 1} a_n n X^{n-1}.</math>

The symbol ''D'' is called the '''formal differentiation operator'''. This definition simply mimics term-by-term differentiation of a polynomial.

This operation is ''R''-[[linear operator|linear]]:

:<math>D(af + bg) = a \cdot Df + b \cdot Dg</math>

for any ''a'', ''b'' in ''R'' and any ''f'', ''g'' in <math>R[[X]].</math> Additionally, the formal derivative has many of the properties of the usual [[derivative]] of calculus. For example, the [[product rule]] is valid:

:<math>D(fg) \ =\ f \cdot (Dg) + (Df) \cdot g,</math>

and the [[chain rule]] works as well:

:<math>D(f\circ g ) = ( Df\circ g ) \cdot Dg,</math>

whenever the appropriate compositions of series are defined (see above under [[#Composition of series|composition of series]]).

Thus, in these respects formal power series behave like [[Taylor series]]. Indeed, for the ''f'' defined above, we find that

:<math>(D^k f)(0) = k! a_k, </math>

where ''D''<sup>''k''</sup> denotes the ''k''th formal derivative (that is, the result of formally differentiating ''k'' times).