Vipul at 21:20, 30 June 2012

2012-06-30T21:20:42Z

← Older revision		Revision as of 21:20, 30 June 2012
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	\| Infinite radius of convergence \|\| If the power series converges absolutely for every choice of nonzero real number <math>x</math>, then the radius of convergence is defined to be <math>\infty</math>. \|\| all of <math>\R</math>.		\| Infinite radius of convergence \|\| If the power series converges absolutely for every choice of nonzero real number <math>x</math>, then the radius of convergence is defined to be <math>\infty</math>. \|\| all of <math>\R</math>.
	\|}		\|}

			==Facts==

			* The radius of convergence of a power series <math>\sum_{k=0}^\infty a_k(x - \mu)^k</math> is given as <math>\frac{1}{\lim \sup_{k \to \infty} \|a_k\|^{1/k}}</math>. For more, see [[rules for determining interval of convergence]].
			* [[Radius of convergence of derivative of power series equals radius of convergence]]

Vipul: /* =For a power series centered at a point other than zero */

2011-09-05T21:15:05Z

=For a power series centered at a point other than zero

← Older revision		Revision as of 21:15, 5 September 2011
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	\|}		\|}

	===For a power series centered at a point other than zero==		===For a power series centered at a point other than zero===

	Consider a power series centered at <math>\mu</math>:		Consider a power series centered at <math>\mu</math>:

Vipul: Created page with "==Definition== ===For a power series centered at zero=== Consider a power series of the form: $\sum_{k=0}^\infty a_kx^k$ We define the radius of convergence using ..."

2011-09-05T21:14:45Z

Created page with "==Definition== ===For a power series centered at zero=== Consider a power series of the form: <math>\sum_{k=0}^\infty a_kx^k</math> We define the radius of convergence using ..."

New page

==Definition==

===For a power series centered at zero===

Consider a power series of the form:

<math>\sum_{k=0}^\infty a_kx^k</math>

We define the radius of convergence using three cases:

{| class="sortable" border="1"
! Case name !! Definition !! Comment about [[interval of convergence]] (points where the power series converges, absolutely or conditionally)
|-
| Finite radius of convergence || The radius of convergence is the largest positive real number <math>c</math>, if it exists, such that the power series is an [[absolutely convergent series]] for all <math>x</math> satisfying <math>|x| < c</math>. || One of these four: <math>[-c,c]</math>, <math>(-c,c)</math>, <math>[-c,c)</math>, and <math>(-c,c]</math>.
|-
| Zero radius of convergence || If there is no nonzero real number <math>x</math> for which the power series converges absolutely, then the radius of convergence is defined to be 0. || <math>\{ 0 \}</math>
|-
| Infinite radius of convergence || If the power series converges absolutely for every choice of nonzero real number <math>x</math>, then the radius of convergence is defined to be <math>\infty</math>. || all of <math>\R</math>.
|}

===For a power series centered at a point other than zero==

Consider a power series centered at <math>\mu</math>:

<math>\sum_{k=0}^\infty a_k(x - \mu)^k</math>

{| class="sortable" border="1"
! Case name !! Definition !! Comment about [[interval of convergence]] (points where the power series converges, absolutely or conditionally)
|-
| Finite radius of convergence || The radius of convergence is the largest positive real number <math>c</math>, if it exists, such that the power series is an [[absolutely convergent series]] for all <math>x</math> satisfying <math>|x - \mu| < c</math>. || One of these four: <math>[\mu-c,\mu + c]</math>, <math>(\mu -c,\mu + c)</math>, <math>[\mu -c,\mu + c)</math>, and <math>(\mu -c,\mu + c]</math>.
|-
| Zero radius of convergence || If there is no nonzero real number <math>x</math> for which the power series converges absolutely, then the radius of convergence is defined to be 0. || <math>\{ \mu \}</math>
|-
| Infinite radius of convergence || If the power series converges absolutely for every choice of nonzero real number <math>x</math>, then the radius of convergence is defined to be <math>\infty</math>. || all of <math>\R</math>.
|}

Radius of convergence - Revision history

Vipul at 21:20, 30 June 2012

Vipul: /* =For a power series centered at a point other than zero */

Vipul: Created page with "==Definition== ===For a power series centered at zero=== Consider a power series of the form: \sum_{k=0}^\infty a_kx^k We define the radius of convergence using ..."

Vipul: Created page with "==Definition== ===For a power series centered at zero=== Consider a power series of the form: $\sum_{k=0}^\infty a_kx^k$ We define the radius of convergence using ..."