Power series

In mathematics, a power series (in one variable) is an infinite series of the form In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant. an is independent of x and may be expressed as a function of n (e.g., a n = 1 / n ! {displaystyle a_{n}=1/n!} ). Power series are useful in analysis since they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form These power series arise primarily in analysis, but also occur in combinatorics as generating functions (a kind of formal power series) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at ​1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series. Any polynomial can be easily expressed as a power series around any center c, although all but finitely many the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial f ( x ) = x 2 + 2 x + 3 { extstyle f(x)=x^{2}+2x+3} can be written as a power series around the center c = 0 { extstyle c=0} as or around the center c = 1 { extstyle c=1} as or indeed around any other center c. One can view power series as being like 'polynomials of infinite degree,' although power series are not polynomials. The geometric series formula which is valid for | x | < 1 { extstyle |x|<1} , is one of the most important examples of a power series, as are the exponential function formula