Overview: A power series is an infinite sum of the form f(x) = a_0 + a_1(x-c) + a_2(x-c)^2 + a_3(x-c)^3 + ⋯, often written f(x) = ∑_{n=0}^∞ a_n (x-c)^n. It is a central object in mathematics, providing a bridge between algebraic expressions and analytic behaviour of functions. The variable x varies near the center c, so a series is said to be centered at c; the special case c = 0 is called a Maclaurin series.

Definition and basic characteristics

Each coefficient a_n is a scalar (real, complex, or in some contexts p-adic) that determines the weight of the nth power. The list of coefficients {a_n} uniquely specifies the power series and, when the series converges, defines a function on the set of x-values where the sum is meaningful. One can treat familiar positional numeration (for example decimal expansions of integers) as a power series with x fixed at 10; in that interpretation, the digits play the role of coefficients and the variable is the base.

Convergence and radius

Unlike finite polynomials, a power series converges only for values of x within a certain distance from the center c. There exists a nonnegative real number R, called the radius of convergence, such that the series converges for |x-c|R; on the boundary |x-c|=R convergence must be checked term by term. The radius R can be computed from the coefficients by the Cauchy–Hadamard formula R = 1 / (limsup_{n→∞} |a_n|^{1/n}). Standard tests (ratio test, root test) are used to decide convergence and to find R in concrete cases. The geometric series ∑ x^n has R=1 and illustrates the boundary behaviour that commonly arises.

Operations and analytic consequences

Within its interval of convergence a power series can be manipulated like a polynomial: it may be added to other series, multiplied (via Cauchy products), and differentiated or integrated termwise, and the result will have the same radius of convergence. This flexibility makes power series a practical tool for solving differential equations and for approximating functions. If a function equals its power series in a neighborhood of a point, it is called analytic there; analytic functions are infinitely differentiable and are determined by their Taylor coefficients.

History, examples, and applications

  • Historically, power series arose from attempts to represent transcendental functions by infinite polynomials; the Taylor series expansion is the standard link between derivatives and coefficients. See the notion of Taylor series for classical examples and the general formula for coefficients.
  • Common examples include the exponential, sine and cosine series, and the geometric series; these are often presented as expansions of a known function.
  • In combinatorics power series are used as generating functions to encode sequences; in electrical engineering notation the Z-transform has a closely related form.
  • Number theory uses related ideas: positional integer representations and p-adic expansions can be viewed through a power-series lens.

Notable distinctions and remarks

Not every infinitely differentiable function is equal to its Taylor series (examples of smooth but non-analytic functions exist). Analyticity is a stronger property: a function is analytic at a point precisely when it can be represented there by a convergent power series. When working in different number systems or rings (complex numbers, p-adics, formal power series) the meaning of convergence or of formal manipulation changes: in formal algebra one studies series without worrying about numeric convergence, while in analysis convergence is essential. For more background and technical details see introductions to series in one variable, the role of coefficients, and historical notes on mathematical development.