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| In [[mathematics]], the '''radius of convergence''' of a [[power series]] is the radius of the largest [[disk (mathematics)|disk]] in which the [[power series|series]] [[Convergent series|converge]]s. It is either a non-negative real number or ∞. When it is positive, the power series [[absolute convergence|converges absolutely]] and [[compact convergence|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.
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| ==Definition==
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| For a power series ''ƒ'' defined as:
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| :<math>f(z) = \sum_{n=0}^\infty c_n (z-a)^n, </math>
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| where
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| :''a'' is a [[complex number|complex]] constant, the center of the [[disk (mathematics)|disk]] of convergence,
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| :''c''<sub>''n''</sub> is the ''n''<sup>th</sup> complex coefficient, and
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| :''z'' is a complex variable.
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| The radius of convergence ''r'' is a nonnegative real number or ∞ such that the series converges if
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| :<math>|z-a| < r\,</math>
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| and diverges if
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| :<math>|z-a| > r.\,</math>
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| In other words, the series converges if ''z'' is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |''z'' − ''a''| = ''r'', the behavior of the power series may be complicated, and the series may converge for some values of ''z'' and diverge for others. The radius of convergence is infinite if the series converges for all [[complex number]]s ''z''.
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| ==Finding the radius of convergence==
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| Two cases arise. The first case is theoretical: when you know all the coefficients <math>c_n</math> then you take certain limits and find the precise radius of convergence. The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of convergence.
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| ===Theoretical radius===
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| The radius of convergence can be found by applying the [[root test]] to the terms of the series. The root test uses the number
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| :<math>C = \limsup_{n\rightarrow\infty}\sqrt[n]{|c_n(z-a)^n|} = \limsup_{n\rightarrow\infty}\sqrt[n]{|c_n|}|z-a|</math>
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| "lim sup" denotes the [[limit superior]]. The root test states that the series converges if ''C'' < 1 and diverges if ''C'' > 1. It follows that the power series converges if the distance from ''z'' to the center ''a'' is less than
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| :<math>r = \frac{1}{\limsup_{n\rightarrow\infty}\sqrt[n]{|c_n|}}</math>
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| and diverges if the distance exceeds that number; this statement is the [[Cauchy–Hadamard theorem]]. Note that ''r'' = 1/0 is interpreted as an infinite radius, meaning that ''ƒ'' is an [[entire function]].
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| The limit involved in the [[ratio test]] is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.
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| <!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_{n+1}/c_n. -->
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| :<math>r = \lim_{n\rightarrow\infty} \left| \frac{c_n}{c_{n+1}} \right|.</math>
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| <!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_n/c_{n+1}. -->
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| This is shown as follows. The ratio test says the series converges if
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| : <math> \lim_{n\to\infty} \frac{|c_{n+1}(z-a)^{n+1}|}{|c_n(z-a)^n|} < 1. </math>
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| That is equivalent to
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| : <math> |z - a| < \frac{1}{\lim_{n\to\infty} \frac{|c_{n+1}|}{|c_n|}} = \lim_{n\to\infty} \left|\frac{c_n}{c_{n+1}}\right|. </math>
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| ==={{anchor|Domb–Sykes plot|Domb-Sykes plot}} Practical estimation of radius=== <!-- [[Domb–Sykes plot]] redirects here (and so in [[MOS:BOLD|boldface]] -->
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| [[File:Domb Sykes plot Hinch.svg|thumb|right|400px|Domb–Sykes plot of the function <math>f(\varepsilon)=\varepsilon\,(1+\varepsilon^3)\,/\,\sqrt{(1+2\varepsilon)}.</math><ref>See Figure 8.1 in: {{citation| first=E.J. |last=Hinch |year=1991 |title=Perturbation Methods |series=Cambridge Texts in Applied Mathematics |volume=6 |publisher=Cambridge University Press |isbn=0-521-37897-4 |page=146}}</ref> On the left (a) is a straightforward plot of the ratio of the power-series coefficients <math>c_{n-1}/c_{n}</math> as a function of index <math>n</math>; on the right, (b) is the Domb–Sykes plot of <math>c_{n}/c_{n-1}</math> as a function of <math>1/n</math>. The solid green line is the [[straight line|straight-line]] [[asymptote]] in the Domb–Sykes plot, and intercepts the vertical axis at −2 and has a slope +1. So there is a singularity at <math>\varepsilon=-\tfrac12</math> and the radius of convergence is <math>r=\tfrac12.</math>]]
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| Suppose you only know a finite number of coefficients <math>c_n</math>, say ten to a hundred. Typically, as <math>n</math> increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity.
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| When the behavior of the coefficients is one of constant sign or alternating sign, [[Cyril Domb|Domb]] and Sykes<ref>{{citation |author1-link=Cyril Domb |first1=C. |last1=Domb |first2=M.F. |last2=Sykes |title=On the susceptibility of a ferromagnet above the Curie point |journal=Proc. Roy. Soc. Lond. A |volume=240 |year=1957 |pages=214–228 |doi=10.1098/rspa.1957.0078 |issue=1221}}</ref> proposed plotting <math>c_n/c_{n-1}</math> against <math>1/n</math>, fitting a straight line extrapolation, and taking the intercept of this line as an estimate the reciprocal <math>1/r</math> of the radius of convergence. Negative <math>r</math> means the convergence-limiting singularity is on the negative axis. This procedure is called a '''Domb–Sykes plot'''.
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| When the coefficients settle into having a periodic pattern of signs then use a test proposed by Mercer and Roberts.<ref>{{citation |first1=G.N. |last1=Mercer |first2=A.J. |last2=Roberts |title=A centre manifold description of contaminant dispersion in channels with varying flow properties |journal=SIAM J. Appl. Math. |volume=50 |pages=1547–1565 |year=1990 |doi=10.1137/0150091 |issue=6}}</ref> Compute <math>b_n</math> from <math>b_n^2=(c_{n+1}c_{n-1}-c_n^2)/(c_nc_{n-2}-c_{n-1}^2)</math> and plot <math>b_n</math> versus <math>1/n</math>. Extrapolate to <math>1/n=0</math> to again estimate the reciprocal <math>1/r</math> of the radius of convergence.
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| You may also estimate two subsidiary quantities. Estimate the exponent <math>p</math> of the convergence limiting singularity because the slope of the straight line extrapolation is <math>-(p+1)/r</math>. Estimate the angle <math>\theta</math>, from the real axis, of the convergence limiting singularities by plotting <math>(c_{n-1}b_n/c_n+c_{n+1}/c_n/b_n)/2</math> versus <math>1/n^2</math>. Then extrapolating to <math>1/n^2=0</math> estimates <math>\cos\theta</math>.
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| == Radius of convergence in complex analysis ==
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| A power series with a positive radius of convergence can be made into a [[holomorphic function]] by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:
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| :The radius of convergence of a power series ''f'' centered on a point ''a'' is equal to the distance from ''a'' to the nearest point where ''f'' cannot be defined in a way that makes it holomorphic.
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| The set of all points whose distance to ''a'' is strictly less than the radius of convergence is called the ''disk of convergence''.
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| [[File:TaylorComplexConv.png|thumb|300px|A graph of the functions explained in the text: Approximations in blue, circle of convergence in white]]
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| ''The nearest point'' means the nearest point in the [[complex plane]], not necessarily on the real line, even if the center and all coefficients are real. For example, the function
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| :<math>f(z)=\frac{1}{1+z^2}</math>
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| has no singularities on the real line, since <math>1+z^2</math> has no real roots. Its Taylor series about 0 is given by
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| :<math>\sum_{n=0}^\infty (-1)^n z^{2n}.</math>
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| The root test shows that its radius of convergence is 1. In accordance with this, the function ''ƒ''(''z'') has singularities at ±''i'', which are at a distance 1 from 0.
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| For a proof of this theorem, see [[analyticity of holomorphic functions]].
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| ===A simple example===
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| The arctangent function of [[trigonometry]] can be expanded in a power series familiar to calculus students:
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| :<math>\arctan(z)=z-\frac{z^3}{3}+\frac{z^5}{5}-\frac{z^7}{7}+\cdots .</math>
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| It is easy to apply the root test in this case to find that the radius of convergence is 1.
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| ===A more complicated example===
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| Consider this power series:
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| :<math>\frac{z}{e^z-1}=\sum_{n=0}^\infty \frac{B_n}{n!} z^n </math>
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| where the rational numbers ''B''<sub>''n''</sub> are the [[Bernoulli numbers]]. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At ''z'' = 0, there is in effect no singularity since [[removable singularity|the singularity is removable]]. The only non-removable singularities are therefore located at the ''other'' points where the denominator is zero. We solve
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| :<math>e^z-1=0\,</math>
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| by recalling that if ''z'' = ''x'' + ''iy'' and ''e''<sup> ''iy''</sup> = cos(''y'') + ''i'' sin(''y'') then
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| :<math>e^z = e^x e^{iy} = e^x(\cos(y)+i\sin(y)),\,</math>
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| and then take ''x'' and ''y'' to be real. Since ''y'' is real, the absolute value of cos(''y'') + ''i'' sin(''y'') is necessarily 1. Therefore, the absolute value of ''e''<sup> ''z''</sup> can be 1 only if ''e''<sup> ''x''</sup> is 1; since ''x'' is real, that happens only if ''x'' = 0. Therefore ''z'' is pure imaginary and cos(''y'') + ''i'' sin(''y'') = 1. Since ''y'' is real, that happens only if cos(''y'') = 1 and sin(''y'') = 0, so that ''y'' is an integral multiple of 2π. Consequently the singular points of this function occur at
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| :''z'' = a nonzero integer multiple of 2π''i''.
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| The singularities nearest 0, which is the center of the power series expansion, are at ±2π''i''. The distance from the center to either of those points is 2π, so the radius of convergence is 2π.
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| == Convergence on the boundary ==
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| If the power series is expanded around the point ''a'' and the radius of convergence is {{math|''r''}}, then the set of all points {{math|''z''}} such that {{math|{{mabs|''z'' − ''a''}} {{=}} ''r''}} is a [[circle]] called the ''boundary'' of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily converge absolutely.
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| Example 1: The power series for the function {{math|''ƒ''(''z'') {{=}} 1/(1 − ''z'')}}, expanded around {{math|''z'' {{=}} 0}}, which is simply
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| :<math> \sum_{n=0}^\infty z^n,</math>
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| has radius of convergence 1, and diverges at every point on the boundary.
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| Example 2: The power series for {{math|''g''(''z'') {{=}} −ln(1 − ''z'')}}, expanded around {{math|''z'' {{=}} 0}}, which is
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| :<math> \sum_{n=1}^\infty \frac{1}{n} z^n,</math>
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| has radius of convergence 1, and diverges for {{math|''z'' {{=}} 1}} but converges for all other points on the boundary. The function {{math|''ƒ''(''z'')}} of Example 1 is the [[derivative]] of {{math|''g''(''z'')}}.
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| Example 3: The power series
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| :<math> \sum_{n=1}^\infty \frac{1}{n^2} z^n </math>
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| has radius of convergence 1 and converges everywhere on the boundary absolutely. If {{math|''h''}} is the function represented by this series on the unit disk, then the derivative of ''h''(''z'') is equal to ''g''(''z'')/''z'' with ''g'' of Example 2. It turns out that {{math|''h''(''z'')}} is the [[dilogarithm]] function.
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| Example 4: The power series
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| :<math>\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for }n=\lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n,</math>
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| has radius of convergence 1 and converges [[uniform convergence|uniformly]] on the entire boundary {|''z''| = 1}, but does not [[Absolute convergence|converge absolutely]] on the boundary.<ref>{{citation|last=Sierpiński|first=Wacław|author-link=Wacław Sierpiński|year=1918|title=O szeregu potęgowym który jest zbieżny na całem swem kole zbieżności jednostajnie ale nie bezwzględnie|periodical=Prace matematyka-fizyka|volume=29|pages=263–266}}</ref>
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| ==Comments on rate of convergence==
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| If we expand the function
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| :<math>f(x)=\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x</math>
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| around the point ''x'' = 0, we find out that the radius of convergence of this series is <math>\scriptstyle\infty</math> meaning that this series converges for all complex numbers. However, in applications, one is often interested in the precision of a [[numerical analysis|numerical answer]]. Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer. For example, if we want to calculate ƒ(0.1) = sin(0.1) accurate up to five decimal places, we only need the first two terms of the series. However, if we want the same precision for ''x'' = 1, we must evaluate and sum the first five terms of the series. For ƒ(10), one requires the first 18 terms of the series, and for ƒ(100), we need to evaluate the first 141 terms.
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| So the fastest convergence of a power series expansion is at the center, and as one moves away from the center of convergence, the [[rate of convergence]] slows down until you reach the boundary (if it exists) and cross over, in which case the [[Series (mathematics)|series]] will diverge.
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| ==A graphical example==
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| Consider the function 1/(''z''<sup>2</sup> + 1).
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| This function has poles at ''z'' = ±''i''.
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| As seen in the first example, the radius of convergence of this function's series in powers of (''z'' − 0) is 1, as the distance from 0 to each of those poles is 1.
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| Then the [[Taylor series]] of this function around ''z'' = 0 will only converge if |''z''| < 1, as depicted on the example on the right.
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| ==Abscissa of convergence of a Dirichlet series==
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| An analogous concept is the '''abscissa of convergence of a [[Dirichlet series]]
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| :<math>\sum_{n=1}^\infty {a_n \over n^s}.</math>
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| Such a series converges if the real part of ''s'' is greater than a particular number depending on the coefficients ''a''<sub>''n''</sub>: the [[abscissa]] of convergence.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Brown | first1=James | last2=Churchill | first2=Ruel | title=Complex variables and applications | publisher=[[McGraw-Hill]] | location=New York | isbn=978-0-07-010905-6 | year=1989}}
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| * {{Citation | last1=Stein | first1=Elias | authorlink=Elias M. Stein |last2=Shakarchi | first2=Rami | title=Complex Analysis | publisher=[[Princeton University Press]] | location=Princeton, New Jersey | isbn=0-691-11385-8 | year=2003}}
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| ==External links==
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| *[http://www.lassp.cornell.edu/sethna/Cracks/What_Is_Radius_of_Convergence.html What is radius of convergence?]
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| [[Category:Analytic functions]]
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| [[Category:Convergence (mathematics)]]
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| [[Category:Mathematical physics]]
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| [[Category:Theoretical physics]]
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