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MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
Outline
1 Power Series Representation of a Function
2 Taylor and Maclaurin Series
Outline
1 Power Series Representation of a Function
2 Taylor and Maclaurin Series
Definition
If f (x)=n=0
cn(xa)n for all x I then the power series is called a powerseries representation for f in xa.
We also sayThe power series converges to f (x).The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
Definition
If f (x)=n=0
cn(xa)n for all x I then the power series is called a powerseries representation for f in xa.
We also sayThe power series converges to f (x).The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
Definition
If f (x)=n=0
cn(xa)n for all x I then the power series is called a powerseries representation for f in xa.
We also sayThe power series converges to f (x).The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
Given f (x)=n=0
cn(xa)n
Theorem (Differentiation of a Power Series)
f (x)=n=0
Dx[cn(xa)n
]
Theorem (Integration of a Power Series)f (x)dx =
[ n=0
cn(xa)n dx
]+C
Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx =
ex +ex2
Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
Outline
1 Power Series Representation of a Function
2 Taylor and Maclaurin Series
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n = f (a)+ f (a)(xa)+ f
(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomesn=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n
= f (a)+ f (a)(xa)+ f(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomesn=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n = f (a)+ f (a)(xa)+ f
(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomesn=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n = f (a)+ f (a)(xa)+ f
(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomes
n=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n = f (a)+ f (a)(xa)+ f
(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomesn=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
DefinitionIf f is a function for which f (n)(a) exists for all n N, then the TaylorSeries for f about x = a is
n=0
f (n)(a)
n!(xa)n = f (a)+ f (a)(xa)+ f
(a)2!
(xa)2+ ...
In the special case when a = 0, this series becomesn=0
f (n)(0)
n!xn = f (0)+ f (0)x+ f
(0)2!
x2+ ...
and we call this series the Maclaurin Series for f .
Remark1 A function f (x) for which f (n)(x) exists for all n N is called an
infinitely differentiable function.
2 The Maclaurin Series is a special type of Taylor Series.
Remark1 A function f (x) for which f (n)(x) exists for all n N is called an
infinitely differentiable function.2 The Maclaurin Series is a special type of Taylor Series.
ExampleFind the Taylor Series of f (x)= lnx about x = 1.
ExampleFind the Maclaurin Series for f (x)= cosx.
ExampleFind the Taylor Series of f (x)= 2x about x =1.
ExampleFind the Taylor Series of f (x)= lnx about x = 1.
ExampleFind the Maclaurin Series for f (x)= cosx.
ExampleFind the Taylor Series of f (x)= 2x about x =1.
ExampleFind the Taylor Series of f (x)= lnx about x = 1.
ExampleFind the Maclaurin Series for f (x)= cosx.
ExampleFind the Taylor Series of f (x)= 2x about x =1.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.
Not all infinitely differentiable functions have a power seriesrepresentation.Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, thenthat power series representation must be the same as its Taylor Series.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.Not all infinitely differentiable functions have a power seriesrepresentation.
Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, thenthat power series representation must be the same as its Taylor Series.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.Not all infinitely differentiable functions have a power seriesrepresentation.Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, thenthat power series representation must be the same as its Taylor Series.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.Not all infinitely differentiable functions have a power seriesrepresentation.Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, thenthat power series representation must be the same as its Taylor Series.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.Not all infinitely differentiable functions have a power seriesrepresentation.Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, then
that power series representation must be the same as its Taylor Series.
RemarkAll infinitely differentiable functions have a Taylor Series about x = a.Not all infinitely differentiable functions have a power seriesrepresentation.Thus, the Taylor Series of f about x = a is NOT ALWAYS a powerseries representation for f . That is, generally,
f (x) 6=n=0
f (n)(a)
n!(xa)n
However, if a function has a power series representation in xa, thenthat power series representation must be the same as its Taylor Series.
Examplen=0
xn
n!is a power series representation for ex in x0.
Thus, the Maclaurin
Series of ex isn=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0. Thus, the
Maclaurin Series for1
1x isn=0
xn .
Examplen=0
xn
n!is a power series representation for ex in x0. Thus, the Maclaurin
Series of ex is
n=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0. Thus, the
Maclaurin Series for1
1x isn=0
xn .
Examplen=0
xn
n!is a power series representation for ex in x0. Thus, the Maclaurin
Series of ex isn=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0. Thus, the
Maclaurin Series for1
1x isn=0
xn .
Examplen=0
xn
n!is a power series representation for ex in x0. Thus, the Maclaurin
Series of ex isn=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0.
Thus, the
Maclaurin Series for1
1x isn=0
xn .
Examplen=0
xn
n!is a power series representation for ex in x0. Thus, the Maclaurin
Series of ex isn=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0. Thus, the
Maclaurin Series for1
1x is
n=0
xn .
Examplen=0
xn
n!is a power series representation for ex in x0. Thus, the Maclaurin
Series of ex isn=0
xn
n!.
Examplen=0
xn is a power series representation for1
1x in x0. Thus, the
Maclaurin Series for1
1x isn=0
xn .
What condition must be satisfied by a function so that its Taylor Seriesabout x = a is a PSR for the function in xa?
That is, when is
f (x)=n=0
f (n)(a)
n!(xa)n?
We will take this up in this the next section.
Remarksinx and cosx satisfy the required condition. As a consequence:
cosx =n=0
(1)nx2n(2n)!
and sinx =n=0
(1)nx2n+1(2n+1)!
What condition must be satisfied by a function so that its Taylor Seriesabout x = a is a PSR for the function in xa? That is, when is
f (x)=n=0
f (n)(a)
n!(xa)n?
We will take this up in this the next section.
Remarksinx and cosx satisfy the required condition. As a consequence:
cosx =n=0
(1)nx2n(2n)!
and sinx =n=0
(1)nx2n+1(2n+1)!
What condition must be satisfied by a function so that its Taylor Seriesabout x = a is a PSR for the function in xa? That is, when is
f (x)=n=0
f (n)(a)
n!(xa)n?
We will take this up in this the next section.
Remarksinx and cosx satisfy the required condition. As a consequence:
cosx =n=0
(1)nx2n(2n)!
and sinx =n=0
(1)nx2n+1(2n+1)!
What condition must be satisfied by a function so that its Taylor Seriesabout x = a is a PSR for the function in xa? That is, when is
f (x)=n=0
f (n)(a)
n!(xa)n?
We will take this up in this the next section.
Remarksinx and cosx satisfy the required condition. As a consequence:
cosx =n=0
(1)nx2n(2n)!
and sinx =n=0
(1)nx2n+1(2n+1)!
What condition must be satisfied by a function so that its Taylor Seriesabout x = a is a PSR for the function in xa? That is, when is
f (x)=n=0
f (n)(a)
n!(xa)n?
We will take this up in this the next section.
Remarksinx and cosx satisfy the required condition. As a consequence:
cosx =n=0
(1)nx2n(2n)!
and sinx =n=0
(1)nx2n+1(2n+1)!
You must know the power series representation for the following, which weshall derive in the lecture:
1 f (x)= 11x , where |x| < 1
2 f (x)= ex , where x R3 f (x)= ln(1x), where |x| < 14 f (x)= sinx, where x R5 f (x)= cosx, where x R
AnnouncementChapter 2 Quiz: January 23, ThursdayMidterm Exam: February 3, Monday, 7-9 PM, MBLH, conflict need to signup with respective recit teachers
Power Series Representation of a FunctionTaylor and Maclaurin Series