Power series
In mathematics, a power series (in one variable) is an infinite series of the form
- [math]\displaystyle{ f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c) + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots }[/math]
where an represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). This series usually appears as the Taylor series of some known function; the Taylor series article contains many examples.
In many situations c is equal to zero, for example when considering a Maclaurin series. In those cases, the power series takes the simpler form
- [math]\displaystyle{ f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots. }[/math]
These power series appear primarily in analysis, but also appear in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for integers can also be viewed as an example of a power series, but with the argument x fixed at 10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.
Power Series Media
The exponential function (in blue), and its improving approximation by the sum of the first n + 1 terms of its Maclaurin power series (in red). So n=0 gives f(x) = 1, n=1 f(x) = 1 + x, n=2 f(x)= 1 + x + x^2/2, n=3 f(x)= 1 + x + x^2/2 + x^3/6etcetera.
