Taylor series operator is linear - Revision history

Vipul: /* Proof */

2012-12-22T21:59:43Z

Proof

← Older revision		Revision as of 21:59, 22 December 2012
Line 20:		Line 20:
	==Proof==		==Proof==

			<center>{{#widget:YouTube\|id=H2Rz8ic9dEs}}</center>
	===Proof of additivity===		===Proof of additivity===

Vipul: /* Proof of additivity */

2012-12-22T21:59:11Z

Proof of additivity

← Older revision		Revision as of 21:59, 22 December 2012
Line 24:		Line 24:
	'''Given''': <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions.		'''Given''': <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions.

	'''To prove''': <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the Taylor series of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of math>g</math> at <math>x_0</math>.		'''To prove''': <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the Taylor series of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of <math>g</math> at <math>x_0</math>.

	'''Proof''': By Fact (1), we have:		'''Proof''': By Fact (1), we have:

Vipul at 15:44, 30 June 2012

2012-06-30T15:44:03Z

← Older revision		Revision as of 15:44, 30 June 2012
Line 32:		Line 32:
	with equality holding [[concept of equality conditional to existence of one side\|whenever the right side makes sense]]. Since by assumption, the right side always makes sense, the left side always makes sense, so <math>f + g</math> is infinitely differentiable at <math>x_0</math> and it makes sense to take its Taylor series.		with equality holding [[concept of equality conditional to existence of one side\|whenever the right side makes sense]]. Since by assumption, the right side always makes sense, the left side always makes sense, so <math>f + g</math> is infinitely differentiable at <math>x_0</math> and it makes sense to take its Taylor series.

	~~First, we~~ recall the expressions for the Taylor series:		We recall the expressions for the Taylor series:

	<math>\mbox{Taylor series for } f \mbox{ at } x_0 = \sum_{k=0}^\infty \frac{f^{(k)}(x_0)}{k!}(x - x_0)^k</math>		<math>\mbox{Taylor series for } f \mbox{ at } x_0 = \sum_{k=0}^\infty \frac{f^{(k)}(x_0)}{k!}(x - x_0)^k</math>

Vipul at 15:17, 30 June 2012

2012-06-30T15:17:16Z

← Older revision		Revision as of 15:17, 30 June 2012
Line 8:		Line 8:
	# ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.		# ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.

			==Related facts==

			===Similar facts===

			* [[Taylor series operator is multiplicative]]
			* [[Taylor series operator commutes with differentiation]]
	==Facts used==		==Facts used==

Vipul: /* Additivity and scalar multiples */

2012-06-30T15:16:46Z

Additivity and scalar multiples

← Older revision		Revision as of 15:16, 30 June 2012
Line 5:		Line 5:
	===Additivity and scalar multiples===		===Additivity and scalar multiples===

	# ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions. Then, <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the [[fact about::Taylor series]] of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of math>g</math> at <math>x_0</math>. Note that we add Taylor series formally by adding the coefficients for each power <math>(x - x_0)^k</math>.		# ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions. Then, <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the [[fact about::Taylor series]] of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of <math>g</math> at <math>x_0</math>. Note that we add Taylor series formally by adding the coefficients for each power <math>(x - x_0)^k</math>.
	# ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.		# ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.

Vipul at 15:16, 30 June 2012

2012-06-30T15:16:16Z

@@ Line 5: / Line 5: @@
 ===Additivity and scalar multiples===
-# ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions. Then, <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the [[fact about::Taylor series]] of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of math>g</math> at <math>x_0</math>.
+# ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions. Then, <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the [[fact about::Taylor series]] of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of math>g</math> at <math>x_0</math>. Note that we add Taylor series formally by adding the coefficients for each power <math>(x - x_0)^k</math>.
 # ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.

Vipul: Created page with "{{linear operator statement for functions}} ==Statement== ===Additivity and scalar multiples=== # ''Additivity'': Suppose $f$ and $g$ are functions defin..."

2011-09-07T02:11:17Z

Created page with "{{linear operator statement for functions}} ==Statement== ===Additivity and scalar multiples=== # ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defin..."

New page

{{linear operator statement for functions}}

==Statement==

===Additivity and scalar multiples===

# ''Additivity'': Suppose <math>f</math> and <math>g</math> are functions defined on subsets of reals that are both infinitely differentiable functions at a point <math>x_0</math> which is in the domain of both functions. Then, <math>f + g</math> is infinitely differentiable at <math>x_0</math> and the [[fact about::Taylor series]] of <math>f + g</math> about <math>x_0</math> is the sum of the Taylor series of <math>f</math> at <math>x_0</math> and the Taylor series of math>g</math> at <math>x_0</math>.
# ''Scalar multiples'': Suppose <math>f</math> is a function defined on a subset of the reals and it is infinitely differentiable at a point <math>x_0</math> in its domain. Suppose <math>\lambda</math> is a real number. Then, <math>\lambda f</math> is infinitely differentiable at <math>x_0</math> and the [[Taylor series]] of the function <math>\lambda f</math> at <math>x_0</math> is <math>\lambda</math> times the Taylor series of <math>f</math> at <math>x_0</math>.