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| {{Calculus |Series}}
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| In [[mathematics]], a '''power series''' (in one variable) is an [[infinite series]] of the form
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| :<math>f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c)^1</math><math> + a_2 (x-c)^2 + a_3 (x-c)^3 + \cdots</math>
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| where ''a<sub>n</sub>'' represents the coefficient of the ''n''th term, ''c'' is a constant, and ''x'' varies around ''c'' (for this reason one sometimes speaks of the series as being ''centered'' at ''c''). This series usually arises as the [[Taylor series]] of some known [[function (mathematics)|function]].
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| In many situations ''c'' is equal to zero, for instance when considering a [[Maclaurin series]]. In such cases, the power series takes the simpler form
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| :<math>
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| f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots.
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| </math>
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| These power series arise primarily in [[Mathematical analysis|analysis]], but also occur in [[combinatorics]] (as [[generating function]]s, a kind of [[formal power series]]) and in electrical engineering (under the name of the [[Z-transform]]). The familiar [[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]]s is also closely related to that of a power series. | |
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| ==Examples==<!-- This section is linked from [[Complex plane]] -->
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| [[Image:Exp series.gif|right|thumb|The [[exponential function]] (in blue), and the sum of the first ''n''+1 terms of its [[Maclaurin series|Maclaurin power series]] (in red).]]
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| Any [[polynomial]] can be easily expressed as a power series around any center ''c'', although most of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial <math>f(x) = x^2 + 2x + 3</math> can be written as a power series around the center <math>c=0</math> as
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| ::<math>f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \cdots \,</math>
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| or around the center <math>c=1</math> as
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| ::<math>f(x) = 6 + 4 (x-1) + 1(x-1)^2</math><math> + 0(x-1)^3 + 0(x-1)^4 + \cdots \,</math>
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| 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.
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| The [[geometric series]] formula
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| ::<math> \frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots,</math>
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| which is valid for <math>|x|<1</math>, is one of the most important examples of a power series, as are the exponential function
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| formula
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| ::<math> e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots,</math>
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| and the sine formula | |
| ::<math> \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>
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| valid for all real x.
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| These power series are also examples of [[Taylor series]]. | |
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| Negative powers are not permitted in a power series, for instance <math>1 + x^{-1} + x^{-2} + \cdots</math>
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| is not considered a power series (although it is a [[Laurent series]]). Similarly, fractional powers such as <math>x^{1/2}</math> are not permitted (but see [[Puiseux series]]). The coefficients <math>a_n</math> are not allowed to depend on <math>x</math>, thus for instance:
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| :<math>\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \cdots \,</math> is not a power series.
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| ==Radius of convergence==
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| A power series will converge for some values of the variable ''x'' and may diverge for others. All power series ''f''(''x'') in powers of (''x''-''c'') will converge at ''x'' = ''c''. (The correct value ''f''(''c'') = ''a''<sub>0</sub> requires interpreting the expression 0<sup>0</sup> as equal to 1.) If ''c'' is not the only convergent point, 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
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| :<math>r=\liminf_{n\to\infty} \left|a_n\right|^{-\frac{1}{n}}</math>
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| or, equivalently,
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| <math>r^{-1}=\limsup_{n\to\infty} \left|a_n\right|^{\frac{1}{n}}</math>
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| (this is the [[Cauchy–Hadamard theorem]]; see [[limit superior and limit inferior]] for an explanation of the notation). A fast way to compute it is
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| :<math>r^{-1}=\lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|</math>
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| if this limit exists.
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| The series [[absolute convergence|converges absolutely]] for |''x'' − ''c''| < ''r'' and [[uniform convergence|converges uniformly]] on every [[Compact space|compact]] [[subset]] of {''x'' : |''x'' − ''c''| < ''r''}. That is, the series is absolutely and [[compactly convergent]] on the interior of the disc of convergence.
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| For |''x'' − ''c''| = ''r'', we cannot make any general statement on whether the series converges or diverges. However, for the case of real variables, [[Abel's theorem]] states that the sum of the series is continuous at ''x'' if the series converges at ''x''. In the case of complex variables, we can only claim continuity along the line segment starting at ''c'' and ending at ''x''.
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| == Operations on power series ==
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| === Addition and subtraction ===
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| 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:
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| :<math>f(x) = \sum_{n=0}^\infty a_n (x-c)^n</math>
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| :<math>g(x) = \sum_{n=0}^\infty b_n (x-c)^n</math>
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| then
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| :<math>f(x)\pm g(x) = \sum_{n=0}^\infty (a_n \pm b_n) (x-c)^n.</math>
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| === Multiplication and division ===
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| With the same definitions above, for the power series of the product and quotient of the functions can be obtained as follows:
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| :<math> f(x)g(x) = \left(\sum_{n=0}^\infty a_n (x-c)^n\right)\left(\sum_{n=0}^\infty b_n (x-c)^n\right)</math>
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| :<math> = \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-c)^{i+j}</math>
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| :<math> = \sum_{n=0}^\infty \left(\sum_{i=0}^n a_i b_{n-i}\right) (x-c)^n.</math>
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| The sequence <math>m_n = \sum_{i=0}^n a_i b_{n-i}</math> is known as the [[convolution]] of the sequences <math>a_n</math> and <math>b_n</math>.
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| For division, observe:
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| :<math> {f(x)\over g(x)} = {\sum_{n=0}^\infty a_n (x-c)^n\over\sum_{n=0}^\infty b_n (x-c)^n} = \sum_{n=0}^\infty d_n (x-c)^n</math>
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| :<math> f(x) = \left(\sum_{n=0}^\infty b_n (x-c)^n\right)\left(\sum_{n=0}^\infty d_n (x-c)^n\right)</math>
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| and then use the above, comparing coefficients.
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| === Differentiation and integration===
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| Once a function is given as a power series, it is [[derivative|differentiable]] on the [[interior (topology)|interior]] of the domain of convergence. It can be [[derivative|differentiated]] and [[integral|integrated]] quite easily, by treating every term separately:
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| ::<math>
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| f^\prime (x) = \sum_{n=1}^\infty a_n n \left( x-c \right)^{n-1}= \sum_{n=0}^\infty a_{n+1} \left(n+1 \right) \left( x-c \right)^{n}
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| </math>
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| ::<math>
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| \int f(x)\,dx = \sum_{n=0}^\infty \frac{a_n \left( x-c \right)^{n+1}} {n+1} + k = \sum_{n=1}^\infty \frac{a_{n-1} \left( x-c \right)^{n}} {n} + k.
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| </math>
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| Both of these series have the same radius of convergence as the original one.
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| == Analytic functions ==
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| A function ''f'' defined on some [[open set|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 (topology)|neighborhood]] ''V'' ⊆ ''U'', such that there exists a power series with center ''a'' which converges to ''f''(''x'') for every ''x'' ∈ ''V''.
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| 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.
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| If a function is analytic, then it is infinitely often 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
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| ::<math>
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| a_n = \frac {f^{\left( n \right)}\left( c \right)} {n!}
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| </math>
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| 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]].
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| 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 ''c''∈''U'' such that ''f''<sup> (''n'')</sup>(''c'') = ''g''<sup> (''n'')</sup>(''c'') for all ''n'' ≥ 0, then ''f''(''x'') = ''g''(''x'') for all ''x'' ∈ ''U''.
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| If a power series with radius of convergence ''r'' is given, one can consider [[analytic continuation]]s 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''.
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| The power series expansion of the [[inverse function]] of an analytic function can be determined using the [[Lagrange inversion theorem]].
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| == Formal power series ==
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| {{main|Formal power series}}
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| 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]].
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| == Power series in several variables ==
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| 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
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| ::<math>
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| f(x_1,\dots,x_n) = \sum_{j_1,\dots,j_n = 0}^{\infty}a_{j_1,\dots,j_n} \prod_{k=1}^n \left(x_k - c_k \right)^{j_k},
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| </math>
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| where ''j'' = (''j''<sub>1</sub>, ..., ''j''<sub>''n''</sub>) is a vector of natural numbers, the coefficients
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| ''a''<sub>(''j<sub>1</sub>,...,j<sub>n</sub>'')</sub> are usually real or complex numbers, and the center ''c'' = (''c''<sub>1</sub>, ..., ''c''<sub>''n''</sub>) and argument ''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) are usually real or complex vectors. In the more convenient [[multi-index]] notation this can be written
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| ::<math>
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| f(x) = \sum_{\alpha \in \mathbb{N}^n} a_{\alpha} \left(x - c \right)^{\alpha}.
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| </math>
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| The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. For instance, the power series <math> \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.
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| == Order of a power series ==
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| Let α be a multi-index for a power series ''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 |α| such that ''a''<sub>α</sub> ≠ 0, or 0 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]].
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| == See also ==
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| * [[Flat function]]
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| * [[Linear approximation]]
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| * [[Random variable]]
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| ==References==
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| *{{SpringerEOM|title=Power series|id=Power_series&oldid=15309|last=Solomentsev|first=E.D.}}
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| == External links ==
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| * {{MathWorld | urlname= FormalPowerSeries | title= Formal Power Series }}
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| * {{MathWorld | urlname= PowerSeries | title= Power Series }}
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| * [http://math.fullerton.edu/mathews/c2003/ComplexPowerSeriesMod.html Complex Power Series Module by John H. Mathews]
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| * [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:Real analysis]]
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| [[Category:Complex analysis]]
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| [[Category:Multivariable calculus]]
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| [[Category:Mathematical series]]
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