In analysis, the radius of convergence is a property of a power series of the form
-
,
which indicates in which area of the real line or the complex plane convergence is guaranteed for the power series .
definition
The radius of convergence is defined as the supremum of all numbers for which the power series for (at least) one converges with :
![\ rho \ geq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e1760b2df774fbfdc7e308dbee01418c4de105)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![r: = \ sup \ left \ {| x-x _ {{0}} | \ \ left | \ \ sum _ {{n = 0}} ^ {{\ infty}} a _ {{n}} (x- x _ {{0}}) ^ {{n}} \ {\ text {converged}} \ right. \ right \}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44759b7d60bb035e5e462cbfba501d583a2aa1cc)
If the power series for all real numbers and all over the complex plane converges, so this amount of unlimited (up) is, they say, the radius of convergence is infinite: .
![r = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/d38c8164ddb69351cdab28da290255fde3b846d4)
Conclusions from the radius of convergence
For a power series with a radius of convergence :
![r> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452)
- If , then the power series is absolutely convergent . At , the series converges at a superlinear convergence speed ; at for with a linear convergence rate of the convergence rate .
![| x-x_ {0} | <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/c80c69bae62415be6d56a6147a18265d292dbaf7)
![r = \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/d38c8164ddb69351cdab28da290255fde3b846d4)
![r <\ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/64446ff4bad5ef2a5517891452e8c091673a340e)
![x \ neq x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59cd73ce908e9de3a7b84ffb2ef92312d8b01e91)
- If , no general statement can be made, but Abelian limit value theorem helps in some situations . If the series converges, it converges in a sub-linear manner .
![| x-x_ {0} | = r](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3dbe6981780252284cb6efca6cf34df4a88ad71)
- If , then the power series is divergent .
![| x-x_ {0} |> r](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7b5afc6a909974e78cffee7e1c36b2bc2afd7e)
If a real power series is considered, the coefficients of which are real numbers, and are also real, then the convergence area after solving the absolute inequalities is the interval and possibly one of the or both edge points. For power series in the complex, that is, all these quantities can be complex numbers, the convergence area of this function series consists of the interior of the circular disk around the center and with radius , the convergence circle , and possibly some of its edge points.
![on}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31)
![x, ~ x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0bee99b29ab920ba41ab50f5ddee44145947ee)
![(x_ {0} -r, ~ x_ {0} + r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/84bb738ded31166ba898f472cb627e7ccada7545)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
In addition, it holds for all that the power series converges equally for all with . There is always a uniform convergence on an inner circle or sub-interval .
![r ^ {{\ prime}} <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/157dd6fd8573663fc924dcf09b9e2456fd432d27)
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![| x-x_ {0} | \ leq r ^ {{\ prime}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9367c5f4e35996fc15ccd8537b8befb91b61cc0c)
Determination of the radius of convergence
The radius of convergence can be calculated using the Cauchy-Hadamard formula: It applies
![r = {\ frac {1} {\ limsup \ limits _ {{n \ rightarrow \ infty}} \ left ({\ sqrt [{n}] {| a_ {n} |}} \ right)}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6218c2232079e3fe6df099f7dc1b293a316494)
The following applies if the Limes superior in the denominator is the same and if it is the same .
![r = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89)
![+ \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831)
![r = + \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/778e9e80cd61f103fa742e9d520949d47f801069)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
If from a certain index onwards all are different from 0 and the following limit exists or is infinite, then the radius of convergence can be more easily through
![on}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31)
![r = \ lim _ {{n \ rightarrow \ infty}} {\ bigg |} {\ frac {a _ {{n}}} {a _ {{n + 1}}}} {\ bigg |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56faa7f8e6522ed9ade3761756a281fe7b470a40)
be calculated. However, this formula is not always applicable, for example for the coefficient sequence: The corresponding series has the radius of convergence 1, but the specified limit does not exist. The Cauchy-Hadamard formula , on the other hand, is always applicable.
![a _ {{2n}} = 1, ~ a _ {{2n + 1}} = 1 / n](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c80cb98856f234a6ad5d66dffbcc88f43eb599)
Examples of different edge behavior
The following three examples of real power series each have a radius of convergence 1, so they converge for all in the interval ; however, the behavior at the edge points is different:
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![(-1.1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e120a3bd60fc89b495dd7ef6039465b7e6a703b1)
-
does not converge at any of the edge points .![\ pm 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0)
-
converges at both edge points and .![-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac)
![+1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958)
-
does not converge at the right edge point ( harmonic series ), but does converge at the left edge point (alternating harmonic series).![+1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958)
![-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac)
Influence of the development point on the radius of convergence
The development point of a power series has a direct influence on the coefficient sequence and thus also on the radius of convergence. For example, consider the analytical function![z_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e72d1d86e86355892b39b8eb32b964834e113bf)
![{\ displaystyle f \ colon \ mathbb {C} \ setminus \ {2 \ mathrm {i} \} \ to \ mathbb {C}, \ z \ mapsto {\ frac {1} {2 \ mathrm {i} -z }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f37532363a2d9af111e081fc455fd60cfb34b48)
in their power series representation
-
.
These transformations follow directly by means of the geometric series . This representation corresponds to the power series around the development point and with the root criterion follows for the radius of convergence .
![z_0 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6015a6c1d9f7912952d18c0ddc01d7a4b901483)
![r_ {0} = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/f53528e2d46da1455983053bfeb0211ab0227aaa)
If, on the other hand, one chooses
as the development point, it follows with some algebraic transformations
![z_ {1} = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e416f0c8ebae00d2c9d0451ffd356d6ec0d246)
-
.
Here, too, the radius of convergence follows using the root criterion .
![r_ {1} = {\ sqrt {8}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80deb357a8dacf8482bd07601b1e4cd8c0fac03f)
A third development point delivers with an analogous procedure
![{\ displaystyle z_ {2} = 3 \ mathrm {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc870f5f3275c60fc234686acbf4b428cc0d13d)
![{\ displaystyle f (z) = {\ frac {1} {- (z-3 \ mathrm {i}) -3 \ mathrm {i} +2 \ mathrm {i}}} = {\ frac {1} { - \ mathrm {i}}} \ cdot {\ frac {1} {1 - {\ bigl (} {\ frac {z-3 \ mathrm {i}} {\ mathrm {i}}} {\ bigr)} }} = {\ frac {1} {- \ mathrm {i}}} \ cdot \ sum _ {n = 0} ^ {\ infty} \ left ({\ frac {z-3 \ mathrm {i}} { \ mathrm {i}}} \ right) ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0665e8486989bb7bdb05428c354f104ae67d6fc0)
as a power series representation with the radius of convergence . If you draw these three radii of convergence around their development points, they all intersect at the point because the function here has a singularity and is not defined. The circle of convergence clearly expands around a development point until it reaches an undefined point of the function.
![r_ {2} = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/c86192d6c9c9817c3846d26a481e10b077b8245c)
![{\ displaystyle z '= 2 \ mathrm {i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b50008a1e13ee03bbcab6733ff3cc79857e355c)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
Derivation
The formulas for the radius of convergence can be derived from the convergence criteria for series .
Root criterion
Cauchy-Hadamard's formula results from the root criterion . According to this criterion, the power series converges
![\ sum \ limits _ {{n = 0}} ^ {{\ infty}} {a_ {n}} \ left (x-x_ {0} \ right) ^ {n},](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad78c70bfcf423ca8173978905e6ec50c6fa218)
absolutely if
![{\ underset {n \ to \ infty} {{\ mathop {\ lim \ sup}}}} \, {\ sqrt [{n}] {\ left | a_ {n} \ left (x-x_ {0} \ right) ^ {n} \ right |}} = \ left | x-x_ {0} \ right | {\ underset {n \ to \ infty} {{\ mathop {\ lim \ sup}}}} \, {\ sqrt [{n}] {| a_ {n} |}} <1.](https://wikimedia.org/api/rest_v1/media/math/render/svg/208fe53af4f07a2de7319e946c0ad57cb3e9fb63)
Solving for returns the radius of convergence
![\ left | x-x_ {0} \ right |](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c68b1d13d2a543c35821a7af6f7fb709693b7b)
![\ left | x-x _ {{0}} \ right | <{\ frac {1} {{\ underset {n \ to \ infty} {{\ mathop {\ lim \ sup}}}} {\ sqrt [{ n}] {| a _ {{n}} |}}}} = \ liminf _ {{n \ rightarrow \ infty}} \, | a _ {{n}} | ^ {{- 1 / n}} = r .](https://wikimedia.org/api/rest_v1/media/math/render/svg/945cfe3b0f417015b40029cba398c4fdbddce720)
Quotient criterion
If almost all of them are not equal to zero, the power series converges according to the quotient criterion if the following condition is met:
![on}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31)
![\ sum \ limits _ {{n = 0}} ^ {{\ infty}} {a_ {n}} \ left (x-x_ {0} \ right) ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4be662440b26e0fe4d90a16193cbd869dbfef76)
![\ limsup _ {{n \ rightarrow \ infty}} \ left | {\ frac {a _ {{n + 1}} \ left (x-x_ {0} \ right) ^ {{n + 1}}} {a_ {n} \ left (x-x_ {0} \ right) ^ {n}}} \ right | = \ limsup _ {{n \ rightarrow \ infty}} \ left | {\ frac {a _ {{n + 1 }}} {a_ {n}}} \ left (x-x_ {0} \ right) \ right | = \ left | x-x_ {0} \ right | \ limsup _ {{n \ rightarrow \ infty}} \ left | {\ frac {a _ {{n + 1}}} {a_ {n}}} \ right | <1.](https://wikimedia.org/api/rest_v1/media/math/render/svg/622461dbe2ba221bea34846f0abaf1893af0bb4f)
Solve for delivers:
![\ left | x-x_ {0} \ right |](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c68b1d13d2a543c35821a7af6f7fb709693b7b)
![\ left | x-x _ {{0}} \ right | <{\ frac {1} {{\ underset {n \ to \ infty} {{\ mathop {\ lim \ sup}}}} \ left | {\ frac {a _ {{n + 1}}} {a _ {{n}}}} \ right |}} = \ liminf _ {{n \ rightarrow \ infty}} \ left | {\ frac {a _ {{n} }} {a _ {{n + 1}}}} \ right | =: r.](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1e6734e0e9fb161332da2e4bc2adb2ab745de1)
So the power series converges for . In general, however, this is not the radius of convergence. This is because the quotient criterion is really weaker than the root criterion in the following sense: Is
![\ left | x-x_ {0} \ right | <r](https://wikimedia.org/api/rest_v1/media/math/render/svg/94c3198adffebbc6dc88d633ca8ff8ad8fdc3ff2)
-
,
so in general it cannot be concluded that the series is diverging. The divergence is obtained from
![\ sum \ limits _ {{n = 0}} ^ {\ infty} {b_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7609601c8d2557fb6ffd7c42f2865960c747f1fe)
-
.
Similar to the above, one concludes that the power series for diverges, where
![\ sum \ limits _ {{n = 0}} ^ {{\ infty}} {a_ {n}} \ left (x-x_ {0} \ right) ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4be662440b26e0fe4d90a16193cbd869dbfef76)
![\ left | x-x_ {0} \ right |> R](https://wikimedia.org/api/rest_v1/media/math/render/svg/dec1e1438b2fe856b8a206cc2937a5abb683d6ab)
-
.
In general, one can therefore only say that the radius of convergence lies between and .
![r](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
![R.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
But the implications of this: The existence of follows and in this particular case
![\ lim _ {{n \ to \ infty}} | a _ {{n + 1}} / a_ {n} |](https://wikimedia.org/api/rest_v1/media/math/render/svg/84f33aa0f64058fbec78582db0b98e74d8427041)
![r = R](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a088bb7b6b4be9ff6b0fdb33fc1dd53af91e356)
![r = R = {\ frac {1} {\ lim \ limits _ {{n \ to \ infty}} | a _ {{n + 1}} / a_ {n} |}} = \ lim \ limits _ {{ n \ to \ infty}} \ left | {\ frac {a_ {n}} {a _ {{n + 1}}}} \ right |](https://wikimedia.org/api/rest_v1/media/math/render/svg/d743266890469ce5b12152ee5977aade8456def2)
the desired radius of convergence.
literature
- E. Freitag, R. Busam: Function theory . 2nd Edition. Springer-Verlag, Berlin 1995, ISBN 3-540-58650-4 .
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Harro Heuser : Textbook of Analysis - Part 1 , 6th edition, Teubner 1989, ISBN 3-519-42221-2 , pp. 542-561
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Klaus Jänich : Function Theory - An Introduction . 6th edition, Springer-Verlag, Berlin 2004, ISBN 3540203923 .