Vipul at 21:22, 30 June 2012

2012-06-30T21:22:07Z

← Older revision		Revision as of 21:22, 30 June 2012
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	===Power series centered at a point===		===Power series centered at a point===

	The '''derivative''' of a [[defining ingredient::power series]] centered at a point math>\mu</math> is the power series obtained by diferentiating each term of the power series and adding up. Explicitly, the derivative of the power series:		The '''derivative''' of a [[defining ingredient::power series]] centered at a point <math>\mu</math> is the power series obtained by diferentiating each term of the power series and adding up. Explicitly, the derivative of the power series:

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

Vipul: Created page with "==Definition== ===Power series centered at a point=== The '''derivative''' of a defining ingredient::power series centered at a point math>\mu is the power series ob..."

2011-09-07T11:38:27Z

Created page with "==Definition== ===Power series centered at a point=== The '''derivative''' of a defining ingredient::power series centered at a point math>\mu</math> is the power series ob..."

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==Definition==

===Power series centered at a point===

The '''derivative''' of a [[defining ingredient::power series]] centered at a point math>\mu</math> is the power series obtained by diferentiating each term of the power series and adding up. Explicitly, the derivative of the power series:

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

is the power series

<math>\sum_{k=1}^\infty ka_k(x - \mu)^{k-1}</math>

Rewriting the dummy variable <math>k</math> as <math>k + 1</math>, we can rewrite the above summation as:

<math>\sum_{k=0}^\infty (k+1)a_{k+1} (x - \mu)^k</math>

==Facts==

* [[Differentiation theorem for power series]]: This states that if a power series converges to a function on a given [[interval of convergence]], the derivative of the power series converges to the derivative of the function on the interior of that interval of convergence. It may happen, though, that the power series converges to the function at the boundary of the interval of convergence but the power series for the derivative does not converge.
* [[Taylor series operator commutes with differentiation]]: The [[Taylor series]] of the derivative of a function at a point is the derivative (in the sense of derivative of power series) of the Taylor series.

Derivative of power series - Revision history

Vipul at 21:22, 30 June 2012

Vipul: Created page with "==Definition== ===Power series centered at a point=== The '''derivative''' of a defining ingredient::power series centered at a point math>\mu is the power series ob..."