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