power series

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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a power series (in one variable) is an
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of the form $\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots$ where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in
mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical ...
, where they arise as
Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of
infinitely differentiable function is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...
s. 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 seriesMaclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1 ...
. In such cases, the power series takes the simpler form $\sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \cdots.$ Beyond their role in mathematical analysis, power series also occur in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other area ...
as
generating 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 ordinar ...
s (a kind of
formal power series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
) and in electronic engineering (under the name of the
Z-transform In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
). The familiar
decimal notation The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
for
real number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s can also be viewed as an example of a power series, with integer coefficients, but with the argument ''x'' fixed at . In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777– ...

, the concept of
p-adic number group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analys ...
s is also closely related to that of a power series.

# Examples

The Maclaurin_power_series_(in_red)..html" ;"title="aclaurin_series.html" ;"title="exponential function (in blue), and the sum of the first ''n'' + 1 terms of its Maclaurin series">Maclaurin power series (in red).">aclaurin_series.html" ;"title="exponential function (in blue), and the sum of the first ''n'' + 1 terms of its Maclaurin series">Maclaurin power series (in red). 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 $f(x) = x^2 + 2x + 3$ can be written as a power series around the center $c = 0$ as $f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \cdots$ or around the center $c = 1$ as $f(x) = 6 + 4(x - 1) + 1(x - 1)^2 + 0(x - 1)^3 + 0(x - 1)^4 + \cdots$ 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 In mathematics, a geometric series (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
formula $\frac = \sum_^\infty x^n = 1 + x + x^2 + x^3 + \cdots,$ which is valid for $, x, < 1$, is one of the most important examples of a power series, as are the exponential function formula $e^x = \sum_^\infty \frac = 1 + x + \frac + \frac + \cdots,$ and the sine formula $\sin(x) = \sum_^\infty \frac = x - \frac + \frac - \frac + \cdots,$ valid for all real ''x''. These power series are also examples of
Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
.

## On the set of exponents

Negative powers are not permitted in a power series; for instance, $1 + x^ + x^ + \cdots$ is not considered a power series (although it is a
Laurent series (Holomorphic functions are analytic, analytic). 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 compl ...

). Similarly, fractional powers such as $x^\frac$ are not permitted (but see
Puiseux series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
). The coefficients $a_n$ are not allowed to depend on thus for instance: $\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \cdots$ is not a power series.

A power series $\sum_^\infty a_n(x-c)^n$ is
convergent Convergent is an adjective for things that wikt:converge, converge. It is commonly used in mathematics and may refer to: *Convergent boundary, a type of plate tectonic boundary * Convergent (continued fraction) * Convergent evolution * Convergent s ...
for some values of the variable , which will always include (as usual, $\left(x-c\right)^0$ evaluates as and the sum of the series is thus $a_0$ for ). The series may diverge for other values of . If is not the only point of convergence, then there is always a number with such that the series converges whenever and diverges whenever . The number is called the
radius of convergence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the power series; in general it is given as $r = \liminf_ \left, a_n\^$ or, equivalently, $r^ = \limsup_ \left, a_n\^\frac$ (this is the
Cauchy–Hadamard theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
; see
limit superior and limit inferior In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for an explanation of the notation). The relation $r^ = \lim_\left, \$ is also satisfied, if this limit exists. The set of the
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s such that is called the
disc of convergence In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the series. The series converges absolutely inside its disc of convergence, and converges uniformly on every
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British N ...
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are u ...

of the disc of convergence. For , there is no general statement on the convergence of the series. However,
Abel's theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
states that if the series is convergent for some value such that , then the sum of the series for is the limit of the sum of the series for where is a real variable less than that tends to .

# Operations on power series

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 $f(x) = \sum_^\infty a_n (x - c)^n$ and $g(x) = \sum_^\infty b_n (x - c)^n$ then $f(x) \pm g(x) = \sum_^\infty (a_n \pm b_n) (x - c)^n.$ It is not true that if two power series $\sum_^\infty a_n x^n$ and $\sum_^\infty b_n x^n$ have the same radius of convergence, then $\sum_^\infty \left(a_n + b_n\right) x^n$ also has this radius of convergence. If $a_n = (-1)^n$ and $b_n = (-1)^ \left(1 - \frac\right)$, then both series have the same radius of convergence of 1, but the series $\sum_^\infty \left(a_n + b_n\right) x^n = \sum_^\infty \frac x^n$ has a radius of convergence of 3.

## Multiplication and division

With the same definitions for $f\left(x\right)$ and $g\left(x\right)$, the power series of the product and quotient of the functions can be obtained as follows: $\begin f(x)g(x) &= \left(\sum_^\infty a_n (x-c)^n\right)\left(\sum_^\infty b_n (x - c)^n\right) \\ &= \sum_^\infty \sum_^\infty a_i b_j (x - c)^ \\ &= \sum_^\infty \left(\sum_^n a_i b_\right) (x - c)^n. \end$ The sequence $m_n = \sum_^n a_i b_$ is known as the
convolution In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of the sequences $a_n$ and For division, if one defines the sequence $d_n$ by $\frac = \frac = \sum_^\infty d_n (x - c)^n$ then $f(x) = \left(\sum_^\infty b_n (x - c)^n\right)\left(\sum_^\infty d_n (x - c)^n\right)$ and one can solve recursively for the terms $d_n$ by comparing coefficients. Solving the corresponding equations yields the formulae based on
determinant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of certain matrices of the coefficients of $f\left(x\right)$ and $g\left(x\right)$ $d_0=\frac$ $d_n=\frac \begin a_n &b_1 &b_2 &\cdots&b_n \\ a_&b_0 &b_1 &\cdots&b_\\ a_&0 &b_0 &\cdots&b_\\ \vdots &\vdots&\vdots&\ddots&\vdots \\ a_0 &0 &0 &\cdots&b_0\end$

## Differentiation and integration

Once a function $f\left(x\right)$ is given as a power series as above, it is
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

on the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
of the domain of convergence. It can be and
integrated Integration may refer to: Biology *Modular integration, where different parts in a module have a tendency to vary together *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome ...

quite easily, by treating every term separately: $\begin f'(x) &= \sum_^\infty a_n n (x - c)^ = \sum_^\infty a_ (n + 1) (x - c)^n, \\ \int f(x)\,dx &= \sum_^\infty \frac + k = \sum_^\infty \frac + k. \end$ Both of these series have the same radius of convergence as the original one.

# Analytic functions

A function ''f'' defined on some
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
''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 A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically ...
''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 Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
of its region of convergence. All
holomorphic function A rectangular grid (top) and its image under a conformal map ''f'' (bottom). In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
s 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 $a_n = \frac$ where $f^\left(c\right)$ denotes the ''n''th derivative of ''f'' at ''c'', and $f^\left(c\right) = f\left(c\right)$. This means that every analytic function is locally represented by its
Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. 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 such that for all , then for all . If a power series with radius of convergence ''r'' is given, one can consider
analytic continuation Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...

s of the series, i.e. analytic functions ''f'' which are defined on larger sets than 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with such that no analytic continuation of the series can be defined at . The power series expansion of the
inverse function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of an analytic function can be determined using the
Lagrange inversion theorem In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (numbe ...
.

## Behavior near the boundary

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'': $\sum_^z^n$ has radius of convergence equal to $1$ and diverges at every point of $, z, =1$. Nevertheless, the sum in $, z, <1$ is $\frac$, which is analytic at every point of the plane except for $z=1$. # ''Convergent at some points divergent at others'': $\sum_^(-1)^\frac$ has radius of convergence $1$. It converges for $z=1$, while it diverges for $z=-1$ # ''Absolute convergence at every point of the boundary'': $\sum_^\frac$ has radius of convergence $1$, while it converges absolutely, and uniformly, at every point of $, z, =1$ due to Weierstrass M-test applied with the hyper-harmonic convergent series $\sum_^\frac$. # ''Convergent on the closure of the disc of convergence but not continuous sum'': gave an example of a power series with radius of convergence $1$, convergent at all points with $, z, =1$, 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.

# Formal power series

In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, one attempts to capture the essence of power series without being restricted to the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
s of real and complex numbers, and without the need to talk about convergence. This leads to the concept of
formal power series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, a concept of great utility in
algebraic combinatorics Algebraic combinatorics is an area of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (m ...
.

# Power series in several variables

An extension of the theory is necessary for the purposes of
multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...
. A power series is here defined to be an infinite series of the form $f(x_1, \dots, x_n) = \sum_^\infty a_ \prod_^n (x_k - c_k)^,$ where is a vector of natural numbers, the coefficients are usually real or complex numbers, and the center and argument are usually real or complex vectors. The symbol $\Pi$ is the product symbol, denoting multiplication. In the more convenient
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus Calculus, originally called infinitesimal calcu ...
notation this can be written $f(x) = \sum_ a_\alpha (x - c)^\alpha.$ where $\N$ is the set of
natural number File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...) In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...
s, and so $\N^n$ is the set of ordered ''n''-
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s 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 $\sum_^\infty x_1^n x_2^n$ 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 $\left(\log , x_1, , \log , x_2, \right)$, where $\left(x_1, x_2\right)$ 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.

# Order of a power series

Let be a multi-index for a power series . The order of the power series ''f'' is defined to be the least value $r$ such that there is ''a''''α'' ≠ 0 with $r = , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n$, or $\infty$ 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 (Holomorphic functions are analytic, analytic). 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 compl ...

.

*