Editing Radius of convergence (section)

==Finding the radius of convergence==
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.

===Theoretical radius===

The radius of convergence can be found by applying the [[root test]] to the terms of the series. The root test uses the number

:<math>C = \limsup_{n\to\infty}\sqrt[n]{|c_n(z-a)^n|} = \limsup_{n\to\infty} \left(\sqrt[n]{|c_n|}\right) |z-a|</math>

"lim&nbsp;sup" denotes the [[limit superior]].  The root test states that the series converges if ''C''&nbsp;<&nbsp;1 and diverges if&nbsp;''C''&nbsp;>&nbsp;1.  It follows that the power series converges if the distance from ''z'' to the center ''a'' is less than

:<math>r = \frac{1}{\limsup_{n\to\infty}\sqrt[n]{|c_n|}}</math>

and diverges if the distance exceeds that number; this statement is the [[Cauchy–Hadamard theorem]].  Note that ''r''&nbsp;=&nbsp;1/0 is interpreted as an infinite radius, meaning that ''f'' is an [[entire function]].

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.

<!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_{n+1}/c_n.  -->
:<math>r = \lim_{n\to\infty} \left| \frac{c_{n}}{c_{n+1}} \right|.</math>
<!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_n/c_{n+1}. -->

This is shown as follows.  The ratio test says the series converges if

: <math> \lim_{n\to\infty} \frac{|c_{n+1}(z-a)^{n+1}|}{|c_n(z-a)^n|} < 1. </math>

That is equivalent to

: <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>

==={{anchor|Domb–Sykes plot|Domb–Sykes plot}} Practical estimation of radius in the case of real coefficients ===
<!-- [[Domb–Sykes plot]] redirects here (and so in [[MOS:BOLD|boldface]] -->
[[File:Domb Sykes plot Hinch.svg|thumb|right|400px|Plots of the function <math>f(\varepsilon)=\frac{\varepsilon(1+\varepsilon^3)}{\sqrt{1+2\varepsilon}}.</math> <br>
The solid green line is the [[straight line|straight-line]] [[asymptote]] in the Domb–Sykes plot,<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> plot (b), which intercepts the vertical axis at −2 and has a slope +1. Thus there is a singularity at <math>\varepsilon=-1/2</math> and so the radius of convergence is <math>r=1/2.</math>]]
Usually, in scientific applications, only a finite number of coefficients <math>c_n</math> are known.  Typically, as <math>n</math> increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity.  In this case, two main techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio <math>1/r</math> where ''r'' is the radius of convergence.

* The basic case is when the coefficients ultimately share a common sign or alternate in sign. As pointed out earlier in the article, in many cases the limit <math display="inline">\lim_{n\to \infty} {c_n / c_{n-1}}</math> exists, and in this case <math display="inline">1/r = \lim_{n \to \infty} {c_n / c_{n-1}}</math>. Negative <math>r</math> means the convergence-limiting singularity is on the negative axis. Estimate this limit, by plotting the <math>c_n/c_{n-1}</math> versus <math>1/n</math>, and graphically extrapolate to <math>1/n=0</math> (effectively <math>n=\infty</math>) via a [[Linear Regression|linear fit]].  The intercept with <math>1/n=0</math> estimates the reciprocal of the radius of convergence, <math>1/r</math>.  This plot is called a '''Domb–Sykes plot'''.<ref>{{citation |first1=C. |last1=Domb |first2=M.F. |last2=Sykes |title=On the susceptibility of a ferromagnetic above the Curie point |journal=Proc. R. Soc. Lond. A |volume=240 |pages=214–228 |year=1957 |issue=1221 |doi=10.1098/rspa.1957.0078 |bibcode=1957RSPSA.240..214D |s2cid=119974403 }}</ref>
* The more complicated case is when the signs of the coefficients have a more complex pattern. Mercer and Roberts proposed the following procedure.<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>  Define the associated sequence <math display="block">b_n^2=\frac{c_{n+1}c_{n-1} - c_n^2}{c_n c_{n-2} - c_{n-1}^2} \quad n=3,4,5,\ldots.</math> Plot the finitely many known <math>b_n</math> versus <math>1/n</math>, and graphically extrapolate to <math>1/n=0</math> via a linear fit.  The intercept with <math>1/n=0</math> estimates the reciprocal of the radius of convergence, <math>1/r</math>.{{pb}} This procedure also estimates two other characteristics of the convergence limiting singularity.  Suppose the nearest singularity is of degree <math>p</math> and has angle <math>\pm\theta</math> to the real axis.  Then the slope of the linear fit given above is <math>-(p+1)/r</math>.  Further, plot <math display="inline">\frac{1}{2} \left(\frac{c_{n-1}b_n}{c_n} + \frac{c_{n+1}}{c_n b_n}\right)</math> versus <math display="inline">1/n^2</math>, then a linear fit extrapolated to <math display="inline">1/n^2=0</math> has intercept at <math>\cos\theta</math>.