In mathematics, a power series (in one variable) is an infinite series of the form
∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) + a 2 ( x − c ) 2 + … {\displaystyle \sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)+a_{2}(x-c)^{2}+\dots }
where a n {\displaystyle a_{n}} represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis, where 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, the center c is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form
∑ n = 0 ∞ a n x n = a 0 + a 1 x + a 2 x 2 + … . {\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\dots .}
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power series with only finitely many non-zero terms. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic 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 and 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.
Examples
Polynomial
Every polynomial of degree d can be expressed as a power series around any center c, where all terms of degree higher than d have a coefficient of zero. For instance, the polynomial f ( x ) = x 2 + 2 x + 3 {\textstyle f(x)=x^{2}+2x+3} can be written as a power series around the center c = 0 {\textstyle c=0} as
f ( x ) = 3 + 2 x + 1 x 2 + 0 x 3 + 0 x 4 + ⋯ {\displaystyle f(x)=3+2x+1x^{2}+0x^{3}+0x^{4}+\cdots }
or around the center c = 1 {\textstyle c=1} as
f ( x ) = 6 + 4 ( x − 1 ) + 1 ( x − 1 ) 2 + 0 ( x − 1 ) 3 + 0 ( x − 1 ) 4 + ⋯ . {\displaystyle f(x)=6+4(x-1)+1(x-1)^{2}+0(x-1)^{3}+0(x-1)^{4}+\cdots .}
One can view power series as being like "polynomials of infinite degree", although power series are not polynomials in the strict sense.
Geometric series, exponential function and sine The geometric series formula
1 1 − x = ∑ n = 0 ∞ x n = 1 + x + x 2 + x 3 + ⋯ , {\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+\cdots ,}
which is valid for | x | < 1 {\textstyle |x|<1} , is one of the most important examples of a power series, as are the exponential function formula
… excerpt ends here. Continue reading the full article.
