In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or
infty
For a power series f defined as:
f(z)=
infty | |
\sum | |
n=0 |
cn(z-a)n,
where
The radius of convergence r is a nonnegative real number or
infty
|z-a|<r
and diverges if
|z-a|>r.
Some may prefer an alternative definition, as existence is obvious:
r=\sup\left\{
infty | |
|z-a| \left| \sum | |
n=0 |
n converges | |
c | |
n(z-a) |
\right.\right\}
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 numbers z.[1]
Two cases arise:
cn
The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number
C=\limsupn\toinfty
n|} | |
\sqrt[n]{|c | |
n(z-a) |
=\limsupn\toinfty\left(\sqrt[n]{|cn|}\right)|z-a|
"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
r=
1 | |
\limsupn\toinfty\sqrt[n]{|cn| |
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 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.
r=\limn\toinfty\left|
cn | |
cn+1 |
\right|.
This is shown as follows. The ratio test says the series converges if
\limn\toinfty
|cn+1(z-a)n+1| | ||||||
|
<1.
That is equivalent to
|z-a|<
1 | |||||||||
|
=\limn\toinfty\left|
cn | |
cn+1 |
\right|.
Usually, in scientific applications, only a finite number of coefficients
cn
n
1/r
r
cn/cn-1
1/n
1/n=0
n=infty
1/n=0
1/r
bn
1/n
1/n=0
1/n=0
1/r
p
\pm\theta
-(p+1)/r
\cos\theta
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:
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.
The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence.
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
f(z)= | 1 |
1+z2 |
has no singularities on the real line, since
1+z2
infty | |
\sum | |
n=0 |
(-1)nz2n.
The root test shows that its radius of convergence is 1. In accordance with this, the function f(z) has singularities at ±i, which are at a distance 1 from 0.
For a proof of this theorem, see analyticity of holomorphic functions.
The arctangent function of trigonometry can be expanded in a power series:
\arctan(z)=z- | z3 |
3 |
+
z5 | - | |
5 |
z7 | |
7 |
+ … .
It is easy to apply the root test in this case to find that the radius of convergence is 1.
Consider this power series:
z | |
ez-1 |
infty | |
=\sum | |
n=0 |
Bn | |
n! |
zn
where the rational numbers Bn 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 the singularity is removable. The only non-removable singularities are therefore located at the other points where the denominator is zero. We solve
ez-1=0
by recalling that if and then
ez=exeiy=ex(\cos(y)+i\sin(y)),
and then take x and y to be real. Since y is real, the absolute value of is necessarily 1. Therefore, the absolute value of e can be 1 only if e is 1; since x is real, that happens only if x = 0. Therefore z is purely imaginary and . Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integer multiple of 2. Consequently the singular points of this function occur at
z = a nonzero integer multiple of 2i.
The singularities nearest 0, which is the center of the power series expansion, are at ±2i. The distance from the center to either of those points is 2, so the radius of convergence is 2.
If the power series is expanded around the point a and the radius of convergence is, then the set of all points such that 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.
Example 1: The power series for the function, expanded around, which is simply
infty | |
\sum | |
n=0 |
zn,
Example 2: The power series for, expanded around, which is
infty | |
\sum | |
n=1 |
1 | |
n |
zn,
Example 3: The power series
infty | |
\sum | |
n=1 |
1 | |
n2 |
zn
Example 4: The power series
infty | |
\sum | |
i=1 |
aiziwhereai=
(-1)n-1 | |
2nn |
forn=
n-1 | |
\lfloorlog | |
2(i)\rfloor+1,theuniqueintegerwith2 |
\lei<2n,
If we expand the function
\sinx=
infty | |
\sum | |
n=0 |
(-1)n | |
(2n+1)! |
x2n+1=x-
x3 | |
3! |
+
x5 | |
5! |
- … forallx
around the point x = 0, we find out that the radius of convergence of this series is
infty
So for these particular values 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 will diverge.
An analogous concept is the abscissa of convergence of a Dirichlet series
infty | |
\sum | |
n=1 |
an | |
ns |
.
Such a series converges if the real part of s is greater than a particular number depending on the coefficients an: the abscissa of convergence.