Formal Power Series - Applications

Applications

Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for the Fibonacci numbers, see the article on Examples of generating functions.

One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting. Consider for instance the following elements of Q]:

\sin(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n+1)!} X^{2n+1}
\cos(X) := \sum_{n \ge 0} \frac{(-1)^n} {(2n)!} X^{2n}

Then one can show that

\sin^2(X) + \cos^2(X) = 1 \,

and

\frac{\partial}{\partial X} \sin(X) = \cos(X)

as well as

\sin (X+Y) = \sin(X) \cos(Y) + \cos(X) \sin(Y)\,

(the latter being valid in the ring Q]).

In algebra, the ring K] (where K is a field) is often used as the "standard, most general" complete local ring over K.

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