Editing Power series

{{other uses}}
{{short description|Infinite sum of monomials}}

<!-- {{Calculus|Series}} -->
In [[mathematics]], a '''power series''' (in one [[variable (mathematics)|variable]]) is an [[infinite series]] of the form
<math display="block">\sum_{n=0}^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots</math>
where ''<math>a_n</math>'' represents the [[coefficient]] of the ''n''th 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 function]]s. In fact, [[Borel's lemma|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
<math display="block">\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots.</math>

The [[partial sum]]s of a power series are [[polynomial]]s, 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 function]]s (a kind of [[formal power series]]) and in electronic engineering (under the name of the [[Z-transform]]). The familiar [[Decimal representation|decimal notation]] for [[real number]]s can also be viewed as an example of a power series, with [[integer]] coefficients, but with the argument ''x'' fixed at {{Fraction|1|10}}. In [[number theory]], the concept of [[p-adic number|''p''-adic numbers]] is also closely related to that of a power series.

==Examples==
===Polynomial===
<!-- This section is linked from [[Complex plane]] -->

[[Image:Exp series.gif|right|thumb|The [[exponential function]] (in blue), and its improving approximation by the sum of the first ''n''&nbsp;+&thinsp;1 terms of its [[Maclaurin series|Maclaurin power series]] (in red). So<br> n=0 gives <math>f(x) = 1</math>,<br> n=1 <math>f(x) = 1 + x</math>,<br> n=2 <math>f(x)= 1 + x + x^2/2</math>, <br> n=3 <math>f(x)= 1 + x + x^2/2 + x^3/6</math> etcetera.]]

Every [[polynomial]] of degree {{mvar|d}} can be expressed as a power series around any center {{math|''c''}}, where all terms of degree higher than {{mvar|d}} have a coefficient of zero.<ref>{{cite book|author=Howard Levi|title=Polynomials, Power Series, and Calculus | url=https://books.google.com/books?id=AcI-AAAAIAAJ|year=1967|publisher=Van Nostrand|pages=24|author-link=Howard Levi}}</ref> For instance, the polynomial <math display="inline">f(x) = x^2 + 2x + 3</math> can be written as a power series around the center <math display="inline">c = 0</math> as
<math display="block">f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \cdots</math>
or around the center <math display="inline">c = 1</math> as
<math display="block">f(x) = 6 + 4(x - 1) + 1(x - 1)^2 + 0(x - 1)^3 + 0(x - 1)^4 + \cdots. </math>

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
<math display="block">\frac{1}{1 - x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots,</math>
which is valid for <math display="inline">|x| < 1</math>, is one of the most important examples of a power series, as are the [[exponential function]] formula
<math display="block">e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots</math>
and the [[Taylor_series#Trigonometric_functions|sine formula]]
<math display="block">\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots,</math>
valid for all real ''x''.  These power series are examples of [[Taylor series]] (or, more specifically, of [[Maclaurin series]]).

=== On the set of exponents ===

Negative powers are not permitted in an ordinary power series; for instance, <math display="inline">x^{-1} + 1 + x^{1} + x^{2} + \cdots</math> is not considered a power series (although it is a [[Laurent series]]). Similarly, fractional powers such as <math display="inline">x^\frac{1}{2}</math> are not permitted; fractional powers arise in [[Puiseux series]]. The coefficients <math display="inline"> a_n</math> must not depend on {{nowrap|<math display="inline">x</math>,}} thus for instance <math display="inline">\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \cdots </math> is not a power series.

==Radius of convergence==

A power series <math display="inline"> \sum_{n=0}^\infty a_n(x-c)^n</math> is [[convergent series|convergent]] for some values of the variable {{math|''x''}}, which will always include {{math|1=''x'' = ''c''}} since <math>(x-c)^0 = 1</math> and the sum of the series is thus <math>a_0</math> for {{math|1=''x'' = ''c''}}. The series may [[divergent series|diverge]] for other values of {{mvar|x}}, possibly all of them. If {{math|''c''}} is not the only point of convergence, then there is always a number {{math|''r''}} with {{math|0 < ''r'' ≤ ∞}} such that the series converges whenever {{math|{{abs|''x'' – ''c''}} < ''r''}} and diverges whenever {{math|{{abs|''x'' – ''c''}} > ''r''}}.  The number {{math|''r''}} is called the [[radius of convergence]] of the power series; in general it is given as
<math display="block">r = \liminf_{n\to\infty} \left|a_n\right|^{-\frac{1}{n}}</math>
or, equivalently,
<math display="block">r^{-1} = \limsup_{n\to\infty} \left|a_n\right|^\frac{1}{n}.</math>
This is the [[Cauchy–Hadamard theorem]]; see [[limit superior and limit inferior]] for an explanation of the notation. The relation
<math display="block">r^{-1} = \lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|</math>
is also satisfied, if this limit exists.

The set of the [[complex number]]s such that {{math|{{abs|''x'' – ''c''}} < ''r''}} is called the [[disc of convergence]] of the series. The series [[absolute convergence|converges absolutely]] inside its disc of convergence and it [[uniform convergence|converges uniformly]] on every [[compact space|compact]] [[subset]] of the disc of convergence.

For {{math|1={{abs|''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 {{mvar|z}} such that {{math|1={{abs|''z'' – ''c''}} = ''r''}}, then the sum of the series for {{math|1=''x'' = ''z''}} is the limit of the sum of the series for {{math|1=''x'' = ''c'' + ''t'' (''z'' – ''c'')}} where {{mvar|t}} is a real variable less than {{val|1}} that tends to {{val|1}}.

== Operations on power series ==

=== Addition and subtraction ===
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
<math display="block">f(x) = \sum_{n=0}^\infty a_n (x - c)^n</math> and <math display="block">g(x) = \sum_{n=0}^\infty b_n (x - c)^n</math>
then
<math display="block">f(x) \pm g(x) = \sum_{n=0}^\infty (a_n \pm b_n) (x - c)^n.</math>

The sum of two power series will have a radius of convergence of at least the smaller of the two radii of convergence of the two series,<ref>Erwin Kreyszig, Advanced Engineering Mathematics, 8th ed, page 747</ref> but possibly larger than either of the two. For instance it is not true that if two power series <math display="inline">\sum_{n=0}^\infty a_n x^n</math> and <math display="inline">\sum_{n=0}^\infty b_n x^n</math> have the same radius of convergence, then <math display="inline">\sum_{n=0}^\infty \left(a_n + b_n\right) x^n</math> also has this radius of convergence: if <math display="inline">a_n = (-1)^n</math> and <math display="inline">b_n = (-1)^{n+1} \left(1 - \frac{1}{3^n}\right)</math>, for instance, then both series have the same radius of convergence of 1, but the series <math display="inline">\sum_{n=0}^\infty \left(a_n + b_n\right) x^n = \sum_{n=0}^\infty \frac{(-1)^n}{3^n} x^n</math> has a radius of convergence of 3.

=== Multiplication and division ===
With the same definitions for <math>f(x)</math> and <math>g(x)</math>, the power series of the product and quotient of the functions can be obtained as follows:
<math display="block">\begin{align}
  f(x)g(x) &= \biggl(\sum_{n=0}^\infty a_n (x-c)^n\biggr)\biggl(\sum_{n=0}^\infty b_n (x - c)^n\biggr) \\
           &= \sum_{i=0}^\infty \sum_{j=0}^\infty  a_i b_j (x - c)^{i+j} \\
           &= \sum_{n=0}^\infty \biggl(\sum_{i=0}^n a_i b_{n-i}\biggr) (x - c)^n.
\end{align}</math>

The sequence <math display="inline">m_n = \sum_{i=0}^n a_i b_{n-i}</math> is known as the [[Cauchy product]] of the sequences <math>a_n</math> and {{nowrap|<math>b_n</math>.}}

For division, if one defines the sequence <math>d_n</math> by
<math display="block">\frac{f(x)}{g(x)} = \frac{\sum_{n=0}^\infty a_n (x - c)^n}{\sum_{n=0}^\infty b_n (x - c)^n} = \sum_{n=0}^\infty d_n (x - c)^n</math>
then 
<math display="block">f(x) = \biggl(\sum_{n=0}^\infty b_n (x - c)^n\biggr)\biggl(\sum_{n=0}^\infty d_n (x - c)^n\biggr)</math>
and one can solve recursively for the terms <math>d_n</math> by comparing coefficients.

Solving the corresponding equations yields the formulae based on [[determinant]]s of certain matrices of the coefficients of <math>f(x)</math> and <math>g(x)</math>
<math display="block">d_0=\frac{a_0}{b_0}</math>
<math display="block">d_n=\frac{1}{b_0^{n+1}} \begin{vmatrix}
a_n    &b_1   &b_2   &\cdots&b_n    \\
a_{n-1}&b_0   &b_1   &\cdots&b_{n-1}\\
a_{n-2}&0     &b_0   &\cdots&b_{n-2}\\
\vdots &\vdots&\vdots&\ddots&\vdots \\
a_0    &0     &0     &\cdots&b_0\end{vmatrix}</math>

=== Differentiation and integration===
Once a function <math>f(x)</math> is given as a power series as above, it is [[derivative|differentiable]] on the [[interior (topology)|interior]] of the domain of convergence. It can be [[derivative|differentiated]] and [[integral|integrated]] by treating every term separately since both differentiation and integration are linear transformations of functions:
<math display="block">\begin{align}
    f'(x) &= \sum_{n=1}^\infty a_n n (x - c)^{n-1} = \sum_{n=0}^\infty a_{n+1} (n + 1) (x - c)^n, \\
  \int f(x)\,dx &= \sum_{n=0}^\infty \frac{a_n (x - c)^{n+1}}{n + 1} + k = \sum_{n=1}^\infty \frac{a_{n-1} (x - c)^n}{n} + k.
\end{align}</math>

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

== Analytic functions ==
{{main|Analytic function}}
A function ''f'' defined on some [[open set|open subset]] ''U'' of '''R''' or '''C''' is called [[Analytic function|analytic]] if it is locally given by a convergent power series. This means that every ''a'' ∈ ''U'' has an open [[neighborhood (topology)|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 [[topological interior|interior]] of its region of convergence. All [[holomorphic function]]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''<sub>''n''</sub> can be computed as
<math display="block">a_n = \frac{f^{\left( n \right)} \left( c \right)}{n!}</math>

where <math>f^{(n)}(c)</math> denotes the ''n''th derivative of ''f'' at ''c'', and <math>f^{(0)}(c) = f(c)</math>. 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 [[connectedness|connected]] open set ''U'', and if there exists an element {{math|''c'' ∈ ''U''}} such that {{math|1=''f''{{i sup|(''n'')}}(''c'') = ''g''{{i sup|(''n'')}}(''c'')}} for all {{math|''n'' ≥ 0}}, then {{math|1=''f''(''x'') = ''g''(''x'')}} for all {{math|''x'' ∈ ''U''}}.

If a power series with radius of convergence ''r'' is given, one can consider [[analytic continuation]]s of the series, that is, analytic functions ''f'' which are defined on larger sets than {{math|{{mset| ''x'' | {{abs|''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]] {{mvar|x}} with {{math|1={{abs|''x'' − ''c''}} = ''r''}} such that no analytic continuation of the series can be defined at {{mvar|x}}.

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

=== 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'': <math display="inline">\sum_{n=0}^{\infty}z^n</math> has radius of convergence equal to <math>1</math> and diverges at every point of <math>|z|=1</math>. Nevertheless, the sum in <math>|z|<1</math> is <math display="inline">\frac{1}{1-z}</math>, which is analytic at every point of the plane except for <math>z=1</math>.
# ''Convergent at some points divergent at others'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n}</math> has radius of convergence <math>1</math>. It converges for <math>z=-1</math>, while it diverges for <math>z=1</math>.
# ''Absolute convergence at every point of the boundary'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n^2}</math> has radius of convergence <math>1</math>, while it converges absolutely, and uniformly, at every point of <math>|z|=1</math> due to [[Weierstrass M-test]] applied with the [[Harmonic series (mathematics)#p-series|hyper-harmonic convergent series]] <math display="inline">\sum_{n=1}^{\infty}\frac{1}{n^2}</math>.
# ''Convergent on the closure of the disc of convergence but not continuous sum'': [[Wacław Sierpiński|Sierpiński]] gave an example<ref>{{cite journal|author=Wacław Sierpiński|title=Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue. (French)|journal=Rendiconti del Circolo Matematico di Palermo| url=https://zbmath.org/?q=an:46.1466.03|year=1916|volume=41|publisher=Palermo Rend.|pages=187–190 | doi=10.1007/BF03018294 |jfm=46.1466.03 | s2cid=121218640| author-link=Wacław Sierpiński}}</ref> of a power series with radius of convergence <math>1</math>, convergent at all points with <math>|z|=1</math>, 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]].

== Formal power series ==
{{main|Formal power series}}
In [[abstract algebra]], one attempts to capture the essence of power series without being restricted to the [[field (mathematics)|field]]s 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]].

== Power series in several variables ==
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
<math display="block">f(x_1, \dots, x_n) = \sum_{j_1, \dots, j_n = 0}^\infty a_{j_1, \dots, j_n} \prod_{k=1}^n (x_k - c_k)^{j_k},</math>
where {{math|1=''j'' = (''j''<sub>1</sub>, …, ''j''<sub>''n''</sub>)}} is a vector of natural numbers, the coefficients {{math|''a''<sub>(''j''<sub>1</sub>, …, ''j''<sub>''n''</sub>)</sub>}} are usually real or complex numbers, and the center {{math|1=''c'' = (''c''<sub>1</sub>, …, ''c''<sub>''n''</sub>)}} and argument {{math|1=''x'' = (''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>)}} are usually real or complex vectors. The symbol <math>\Pi</math> is the [[multiplication#Capital Pi notation|product symbol]], denoting multiplication. In the more convenient [[multi-index]] notation this can be written
<math display="block">f(x) = \sum_{\alpha \in \N^n} a_\alpha (x - c)^\alpha.</math>
where <math>\N</math> is the set of [[natural number]]s, and so <math>\N^n</math> is the set of ordered ''n''-[[tuple]]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 <math display="inline">\sum_{n=0}^\infty x_1^n x_2^n</math> is absolutely convergent in the set <math>\{ (x_1, x_2): |x_1 x_2| < 1\}</math> between two hyperbolas.  (This is an example of a ''log-convex set'', in the sense that the set of points <math>(\log |x_1|, \log |x_2|)</math>, where <math>(x_1, x_2)</math> 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.<ref>{{cite journal |doi=10.1090/S0002-9904-1948-08994-7|title=Convex functions|year=1948|last1=Beckenbach|first1=E. F.|journal=Bulletin of the American Mathematical Society|volume=54|issue=5|pages=439–460|doi-access=free}}</ref>

== Order of a power series ==
Let {{mvar|α}} be a multi-index for a power series {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>)}}. The '''order''' of the power series ''f'' is defined to be the least value <math>r</math> such that there is ''a''<sub>''α''</sub> ≠ 0 with <math>r = |\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n</math>, or <math>\infty</math> 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]].

== Notes ==
{{Reflist}}

== References ==
*{{SpringerEOM|title=Power series|id=Power_series&oldid=15309|last=Solomentsev|first=E.D.}}

== External links ==
* {{MathWorld | urlname= FormalPowerSeries | title= Formal Power Series }}
* {{MathWorld | urlname= PowerSeries | title= Power Series }}
* [http://demonstrations.wolfram.com/PowersOfComplexNumbers/ Powers of Complex Numbers] by Michael Schreiber, [[Wolfram Demonstrations Project]].
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{{DEFAULTSORT:Power Series}}
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[[Category:Complex analysis]]
[[Category:Multivariable calculus]]
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