In mathematics, a **power series** (in one variable) is an infinite series of the form

- Examples
- On the set of exponents
- Radius of convergence
- Operations on power series
- Addition and subtraction
- Multiplication and division
- Differentiation and integration
- Analytic functions
- Behavior near the boundary
- Formal power series
- Power series in several variables
- Order of a power series
- Notes
- References
- External links

where *a _{n}* represents the coefficient of the

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

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, 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 of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial can be written as a power series around the center as

or around the center as

or indeed around any other center *c*.^{ [1] } 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 , is one of the most important examples of a power series, as are the exponential function formula

and the sine formula

valid for all real *x*.

These power series are also examples of Taylor series.

Negative powers are not permitted in a power series; for instance, is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as are not permitted (but see Puiseux series). The coefficients are not allowed to depend on , thus for instance:

is not a power series.

A power series is convergent for some values of the variable *x*, which will always include *x* = *c* (as usual, evaluates as 1 and the sum of the series is thus for *x* = *c*). The series may diverge for other values of x. If *c* is not the only point of convergence, then there is always a number *r* with 0 < *r* ≤ ∞ such that the series converges whenever |*x* – *c*| < *r* and diverges whenever |*x* – *c*| > *r*. The number *r* is called the radius of convergence of the power series; in general it is given as

or, equivalently,

(this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). The relation

is also satisfied, if this limit exists.

The set of the complex numbers such that |*x* – *c*| < *r* is called the disc of convergence of the series. The series converges absolutely inside its disc of convergence, and converges uniformly on every compact subset of the disc of convergence.

For |*x* – *c*| = *r*, there is no general statement on the convergence of the series. However, Abel's theorem states that if the series is convergent for some value z such that |*z* – *c*| = *r*, then the sum of the series for *x* = *z* is the limit of the sum of the series for *x* = *c* + *t* (*z* – *c*) where t is a real variable less than 1 that tends to 1.

When two functions *f* and *g* are decomposed into power series around the same center *c*, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if

- and

then

It is not true that if two power series and have the same radius of convergence, then also has this radius of convergence. If and , then both series have the same radius of convergence of 1, but the series has a radius of convergence of 3.

With the same definitions for and , the power series of the product and quotient of the functions can be obtained as follows:

The sequence is known as the convolution of the sequences and .

For division, if one defines the sequence by

then

and one can solve recursively for the terms by comparing coefficients.

Solving the corresponding equations yields the formulae based on determinants of certain matrices of the coefficients of and

Once a function is given as a power series as above, it is differentiable on the interior of the domain of convergence. It can be differentiated and integrated quite easily, by treating every term separately:

Both of these series have the same radius of convergence as the original one.

A function *f* defined on some open subset *U* of **R** or **C** is called analytic if it is locally given by a convergent power series. This means that every *a* ∈ *U* has an open neighborhood *V* ⊆ *U*, such that there exists a power series with center *a* that converges to *f*(*x*) for every *x* ∈ *V*.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.

If a function is analytic, then it is infinitely differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients *a*_{n} can be computed as

where denotes the *n*th derivative of *f* at *c*, and . This means that every analytic function is locally represented by its Taylor series.

The global form of an analytic function is completely determined by its local behavior in the following sense: if *f* and *g* are two analytic functions defined on the same connected open set *U*, and if there exists an element *c*∈*U* such that *f*^{ (n)}(*c*) = *g*^{ (n)}(*c*) for all *n* ≥ 0, then *f*(*x*) = *g*(*x*) for all *x* ∈ *U*.

If a power series with radius of convergence *r* is given, one can consider analytic continuations of the series, i.e. analytic functions *f* which are defined on larger sets than { *x* : |*x* − *c*| < *r* } and agree with the given power series on this set. The number *r* is maximal in the following sense: there always exists a complex number *x* with |*x* − *c*| = *r* such that no analytic continuation of the series can be defined at *x*.

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:

*Divergence while the sum extends to an analytic function*: has radius of convergence equal to and diverges at every point of . Nevertheless, the sum in is , which is analytic at every point of the plane except for .*Convergent at some points divergent at others.*: has radius of convergence . It converges for , while it diverges for*Absolute convergence at every point of the boundary*: has radius of convergence , while it converges absolutely, and uniformly, at every point of due to Weierstrass M-test applied with the hyper-harmonic convergent series .*Convergent on the closure of the disc of convergence but not continuous sum*: Sierpiński gave an example^{ [2] }of a power series with radius of convergence , convergent at all points with , but the sum is an unbounded function and, in particular, discontinuous. A sufficient condition for one-sided continuity at a boundary point is given by Abel's theorem.

In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics.

An extension of the theory is necessary for the purposes of multivariable calculus. A **power series** is here defined to be an infinite series of the form

where *j* = (*j*_{1}, ..., *j*_{n}) is a vector of natural numbers, the coefficients *a*_{(j1, …, jn)} are usually real or complex numbers, and the center *c* = (*c*_{1}, ..., *c*_{n}) and argument *x* = (*x*_{1}, ..., *x*_{n}) are usually real or complex vectors. The symbol is the product symbol, denoting multiplication. In the more convenient multi-index notation this can be written

where is the set of natural numbers, and so is the set of ordered *n*-tuples of natural numbers.

The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. For instance, the power series is absolutely convergent in the set between two hyperbolas. (This is an example of a *log-convex set*, in the sense that the set of points , where lies in the above region, is a convex set. More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this sense.) On the other hand, in the interior of this region of convergence one may differentiate and integrate under the series sign, just as one may with ordinary power series.

Let *α* be a multi-index for a power series *f*(*x*_{1}, *x*_{2}, ..., *x*_{n}). The **order** of the power series *f* is defined to be the least value such that there is *a*_{α} ≠ 0 with , or if *f* ≡ 0. In particular, for a power series *f*(*x*) in a single variable *x*, the order of *f* is the smallest power of *x* with a nonzero coefficient. This definition readily extends to Laurent series.

- ↑ Howard Levi (1967).
*Polynomials, Power Series, and Calculus*. Van Nostrand. p. 24. - ↑ Wacław Sierpiński (1916).
*Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue. (French)*. Palermo Rend. pp. 187–190.

In mathematics, a **series** is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In mathematics, **real analysis** is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematics, the **Taylor series** of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor who introduced them in 1715.

In mathematics, a **Fourier series** is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle of the summation can be made to approximate an arbitrary function in that interval. As such, the summation is a **synthesis** of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier **analysis**. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

In mathematics, a **formal power series** is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring.

In mathematics, the **Laurent series** of a complex function *f*(*z*) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.

In mathematics, the **radius of convergence** of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or . When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

In mathematics, an **analytic function** is a function that is locally given by a convergent power series. There exist both **real analytic functions** and **complex analytic functions**. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about *x*_{0} converges to the function in some neighborhood for every *x*_{0} in its domain.

In mathematics, an infinite series of numbers is said to **converge absolutely** if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to **converge absolutely** if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

In complex analysis, a branch of mathematics, **analytic continuation** is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.

In mathematical analysis, the **Lagrange inversion theorem**, also known as the **Lagrange–Bürmann formula**, gives the Taylor series expansion of the inverse function of an analytic function.

In mathematics, a **generating function** is a way of encoding an infinite sequence of numbers (*a*_{n}) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the *formal* power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers.

In mathematics, a **Dirichlet series** is any series of the form

In mathematics, the **binomial series** is the Taylor series for the function given by where is an arbitrary complex number and |*x*| < 1. Explicitly,

In mathematics, more specifically in mathematical analysis, the **Cauchy product** is the discrete convolution of two infinite series. It is named after the French mathematician Augustin Louis Cauchy.

In mathematics, a **divergent series** is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In mathematics, the **root test** is a criterion for the convergence of an infinite series. It depends on the quantity

In mathematics, **Borel summation** is a summation method for divergent series, introduced by Émile Borel (1899). It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.

In real analysis and complex analysis, a branch of mathematics, the **identity theorem** for analytic functions states: given functions *f* and *g* analytic on a domain *D*, if *f* = *g* on some , where has an accumulation point, then *f* = *g* on *D*.

In mathematics, for a sequence of complex numbers *a*_{1}, *a*_{2}, *a*_{3}, ... the **infinite product**

- Solomentsev, E.D. (2001) [1994], "Power series",
*Encyclopedia of Mathematics*, EMS Press

- Weisstein, Eric W. "Formal Power Series".
*MathWorld*. - Weisstein, Eric W. "Power Series".
*MathWorld*. - Powers of Complex Numbers by Michael Schreiber, Wolfram Demonstrations Project.

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