MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

Outline

1 Taylor and Maclaurin Series

2 Approximation Using Taylor Polynomials

3 Applications of Power Series: Sums of Infinite Series

Outline

1 Taylor and Maclaurin Series

2 Approximation Using Taylor Polynomials

3 Applications of Power Series: Sums of Infinite 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

In the special case when a = 0, this series becomesn=0

f (n)(0)

n!xn

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

In the special case when a = 0, this series becomesn=0

f (n)(0)

n!xn

and we call this series the Maclaurin Series for f .

RemarkThe Taylor Series of f about x = a is NOT ALWAYS a power seriesrepresentation 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 Seriesabout x = a.

RemarkThe Taylor Series of f about x = a is NOT ALWAYS a power seriesrepresentation 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 Seriesabout x = a.

Outline

1 Taylor and Maclaurin Series

2 Approximation Using Taylor Polynomials

3 Applications of Power Series: Sums of Infinite Series

Definition

If f can be differentiated n times at a, then we define the nth TaylorPolynomial for f about x = a to be

pn(x)=n

k=0

f (k)(a)

k !(xa)k

In the special case when a = 0, this polynomial becomes

pn(x)=n

k=0

f (k)(0)

k !xk

and we call this the nth Maclaurin Polynomial for f .

Remark

If f is infinitely differentiable, then its nth Taylor Polynomial is the sum ofthe first n+1 terms of its Taylor Series (up to the term of degree n).

Definition

If f can be differentiated n times at a, then we define the nth TaylorPolynomial for f about x = a to be

pn(x)=n

k=0

f (k)(a)

k !(xa)k

In the special case when a = 0, this polynomial becomes

pn(x)=n

k=0

f (k)(0)

k !xk

and we call this the nth Maclaurin Polynomial for f .

Remark

If f is infinitely differentiable, then its nth Taylor Polynomial is the sum ofthe first n+1 terms of its Taylor Series (up to the term of degree n).

Definition

If f can be differentiated n times at a, then we define the nth TaylorPolynomial for f about x = a to be

pn(x)=n

k=0

f (k)(a)

k !(xa)k

In the special case when a = 0, this polynomial becomes

pn(x)=n

k=0

f (k)(0)

k !xk

and we call this the nth Maclaurin Polynomial for f .

Remark

If f is infinitely differentiable, then its nth Taylor Polynomial is the sum ofthe first n+1 terms of its Taylor Series (up to the term of degree n).

Definition

If f can be differentiated n times at a, then we define the nth TaylorPolynomial for f about x = a to be

pn(x)=n

k=0

f (k)(a)

k !(xa)k

In the special case when a = 0, this polynomial becomes

pn(x)=n

k=0

f (k)(0)

k !xk

and we call this the nth Maclaurin Polynomial for f .

Remark

If f is infinitely differentiable, then its nth Taylor Polynomial is the sum ofthe first n+1 terms of its Taylor Series (up to the term of degree n).

Definition

If f can be differentiated n times at a, then we define the nth TaylorPolynomial for f about x = a to be

pn(x)=n

k=0

f (k)(a)

k !(xa)k

In the special case when a = 0, this polynomial becomes

pn(x)=n

k=0

f (k)(0)

k !xk

and we call this the nth Maclaurin Polynomial for f .

Remark

If f is infinitely differentiable, then its nth Taylor Polynomial is the sum ofthe first n+1 terms of its Taylor Series (up to the term of degree n).

Example

Find the third degree Taylor Polynomial of f (x)=px1 about x = 2.

ExampleFind the fourth degree Maclaurin Polynomial of ex .

Example

Find the third degree Taylor Polynomial of f (x)=px1 about x = 2.

ExampleFind the fourth degree Maclaurin Polynomial of ex .

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a.

Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial.

If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:

1 The higher the degree of the Taylor polynomial, the better theapproximation.

2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.

2 The closer x0 is to a, the better the approximation.

(Illustration)

Approximation using Taylor Polynomials

Suppose f can be differentiated n times at a. Let pn(x) be its nth Taylorpolynomial. If x0 is very close to a, then

f (x0) pn(x0)

Note the following:1 The higher the degree of the Taylor polynomial, the better the

approximation.2 The closer x0 is to a, the better the approximation.

(Illustration)

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625

p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317

3pe = 1.3956124251

Example

If f (x)=px1, then its third-degree Taylor Polynomial about x = 2 is

P3(x)= 1+ 12(x2) 1

8(x2)2+ 1

16(x2)3

Use this to approximatep1.5.

P3(2.5)= 1.2265625p1.5= 1.224744

ExampleApproximate 3

pe using the fourth-degree Maclaurin Polynomial of ex .

P4

(1

3

)= 1.3955761317 3pe = 1.3956124251

NoteFor your information only (that is, this will not come out in the exam):If lim

n f (x)pn(x)= 0, then

the Taylor series of f is a PSR for f , or

f (x)=n=0

f (n)(a)

n!(xa)n .

What must you know about the relationship between Taylor Series andPSR for f ?

NoteFor your information only (that is, this will not come out in the exam):If lim

n f (x)pn(x)= 0, then the Taylor series of f is a PSR for f , or

f (x)=n=0

f (n)(a)

n!(xa)n .

What must you know about the relationship between Taylor Series andPSR for f ?

NoteFor your information only (that is, this will not come out in the exam):If lim

n f (x)pn(x)= 0, then the Taylor series of f is a PSR for f , or

f (x)=n=0

f (n)(a)

n!(xa)n .

What must you know about the relationship between Taylor Series andPSR for f ?

Outline

1 Taylor and Maclaurin Series

2 Approximation Using Taylor Polynomials

3 Applications of Power Series: Sums of Infinite Series

Example

1 Show that1

(1x)2 =n=0

nxn1 (Exercise)

2 Use the above to find the sum of the infinite seriesn=1

n

3n

Example

1 Show that ex/2 =n=0

(1)nxn2nn!

(Exercise)

2 Use the above to show thatn=1

(1)n(ln2)nn!2n+1

=p224

.

Example

1 Show that1

1+x2 =n=0

(1)nx2n (Exercise)2 Use the above to find a PSR for tan1 x.3 Find an infinite series whose sum is

pi

4.

AnnouncementChapter 2 Quiz: January 23, ThursdayMidterm Exam: February 3, Monday, 7-9 AM, MBLH, conflict need tosign up with respective recit teachers

Taylor and Maclaurin SeriesApproximation Using Taylor PolynomialsApplications of Power Series: Sums of Infinite Series