Radius of convergence: Difference between revisions

Latest revision as of 15:41, 12 September 2008

Definition

Over the complex numbers

Consider the power series about a point $z_{0} \in C$ with coefficients $a_{n} \in C$ :

$\sum a_{n} (z - z_{0})^{n}$

The radius of convergence of this power series is defined in the following equivalent ways:

It is defined as the largest $R$ such that the power series converges absolutely for all $z$ with $| z - z_{0} | < R$ and diverges for all $| z - z_{0} | > R$ .
It is given by the formula:

$R = \frac{1}{lim sup {| a_{n} |}^{1 / n}}$

If the denominator is $\infty$ , the radius of convergence is defined as 0, and if the denominator is 0, the radius of convergence is taken to be $\infty$ .

The open disk centered at $z_{0}$ and of radius equal to $R$ is termed the disk of convergence.

Over the real numbers

Consider the power series about a point $x_{0} \in R$ with coefficients $a_{n} \in R$ :

$\sum a_{n} (x - x_{0})^{n}$

The radius of convergence of this power series is defined in the following equivalent ways:

It is defined as the largest $R$ such that the power series converges absolutely for all $x$ with $| x - x_{0} | < R$ and diverges for all $| x - x_{0} | > R$ .
It is given by the formula:

$R = \frac{1}{lim sup {| a_{n} |}^{1 / n}}$

If the denominator is $\infty$ , the radius of convergence is defined as 0, and if the denominator is 0, the radius of convergence is taken to be $\infty$ .

Facts

Power series is infinitely differentiable in disk of convergence: This result holds both over the reals and the complex numbers.

@@ Line 1: / Line 1: @@
 ==Definition==
+===Over the complex numbers===
 Consider the power series about a point <math>z_0 \in \mathbb{C}</math> with coefficients <math>a_n \in \mathbb{C}</math>:
@@ Line 7: / Line 8: @@
 The '''radius of convergence''' of this power series is defined in the following equivalent ways:
+* It is defined as the largest <math>R</math> such that the power series converges absolutely for all <math>z</math> with <math>\left| z - z_0 \right| < R</math> and diverges for all <math>\left|z - z_0 \right| > R</math>.
+* It is given by the formula:
+<math>R = \frac{1}{\lim \sup \left| a_n \right|^{1/n}}</math>
+If the denominator is <math>\infty</math>, the radius of convergence is defined as 0, and if the denominator is 0, the radius of convergence is taken to be <math>\infty</math>.
+The open disk centered at <math>z_0</math> and of radius equal to <math>R</math> is termed the ''disk of convergence''.
+===Over the real numbers===
+Consider the power series about a point <math>x_0 \in \R</math> with coefficients <math>a_n \in \R</math>:
+<math>\sum a_n(x - x_0)^n</math>
+The '''radius of convergence''' of this power series is defined in the following equivalent ways:
+* It is defined as the largest <math>R</math> such that the power series converges absolutely for all <math>x</math> with <math>\left| x - x_0 \right| < R</math> and diverges for all <math>\left|x - x_0 \right| > R</math>.
 * It is given by the formula:
@@ Line 14: / Line 33: @@
 ==Facts==
+* [[Power series is infinitely differentiable in disk of convergence]]: This result holds both over the reals and the complex numbers.