Radius of convergence - Revision history

Vipul at 15:41, 12 September 2008

2008-09-12T15:41:10Z

← Older revision		Revision as of 15:41, 12 September 2008
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	The '''radius of convergence''' of this power series is defined in the following equivalent ways:		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 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:		* It is given by the formula:

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	The '''radius of convergence''' of this power series is defined in the following equivalent ways:		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 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:		* It is given by the formula:

Vipul at 15:39, 12 September 2008

2008-09-12T15:39:55Z

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 ==Definition==
 Consider the power series about a point <math>z_0 \in \mathbb{C}</math> with coefficients <math>a_n \in \mathbb{C}</math>:
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 The '''radius of convergence''' of this power series is defined in the following equivalent ways:
 * It is given by the formula:
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 ==Facts==

Vipul: 1 revision

2008-05-18T19:17:44Z

1 revision

← Older revision	Revision as of 19:17, 18 May 2008
(No difference)

Vipul: New page: ==Definition== Consider the power series about a point $z_0 \in \mathbb{C}$ with coefficients $a_n \in \mathbb{C}$ : $\sum a_n(z - z_0)^n$ The '''radius ...

2008-04-13T19:42:00Z

New page: ==Definition== Consider the power series about a point <math>z_0 \in \mathbb{C}</math> with coefficients <math>a_n \in \mathbb{C}</math>: <math>\sum a_n(z - z_0)^n</math> The '''radius ...

New page

==Definition==

Consider the power series about a point <math>z_0 \in \mathbb{C}</math> with coefficients <math>a_n \in \mathbb{C}</math>:

<math>\sum a_n(z - z_0)^n</math>

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

* 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>.

==Facts==

Radius of convergence - Revision history

Vipul at 15:41, 12 September 2008

Vipul at 15:39, 12 September 2008

Vipul: 1 revision

Vipul: New page: ==Definition== Consider the power series about a point z_0 \in \mathbb{C} with coefficients a_n \in \mathbb{C}: \sum a_n(z - z_0)^n The '''radius ...

Vipul: New page: ==Definition== Consider the power series about a point $z_0 \in \mathbb{C}$ with coefficients $a_n \in \mathbb{C}$ : $\sum a_n(z - z_0)^n$ The '''radius ...