Remainder Theorem

The n-th Talor polynomial

The polynomial

is called the n-th Taylor polynomial for f about c

n

k

kk

n cxk

cfxp

0

)(

)(!

)()(

The n-th Maclaurin polynomial

The polynomial

is called the n-th Maclaurin polynomial for f

n

k

k

nkx

k

fxp

0

)(

!

)0()(

Taylor formula for f with the Remainder

The difference

is called the n-th remainder for the Taylor series of f about c.

n

k

kk

nn

cxk

cfxf

xPxfxR

0

)(

)(!

)()(

)()()(

Thr Remainder Theorem

Assume that all the first n+1 derivatives exist on an interval l containing c and

Ixcxn

xR

Then

numberpositivesomeforIxxfSup

n

n

n

;)!1(

)(

,}:)({

1

)1(

Example (1)Approximate e to five decimal places

kn

k

nn

x

xk

xn

xxxxxp

iseforpolynomialMaclaurinThe

haveWe

0

432

!

1!

1

!4

1

!3

1

!2

11)(

,

1

6

6)1(

6)1(

1

432

0

01)!1(

3)1(

,

10int)10

11,1(

3)}10

11,1(:)(sup{

3

)}10

11,1(:)({

01)!1(

)1(

!

1

!4

1

!3

1

!2

11

!

1)(

n

n

xn

xn

n

n

n

kn

kn

x

nR

Thus

andcontainingervalanisIthatand

xexfthatNotice

examplefor

xexf

settheforboundupperenoughsmallaiswheren

R

saystheoremThe

xn

xxxx

xk

xp

iseforpolynomialMaclaurinThe

71828.2

)1(!9

1)1(

!8

1)1(

!7

1)1(

!6

1)1(

!5

1)1(

!4

1)1(

!3

1)1(

!2

111

:5

?9

600000)!1(

000005.0)!1(

3

000005.0

98765432

e

isplacesdecimaltoaccurateeofionapproximatan

WhynChoose

n

n

Let

needweaccuracyThe

Why we chose n=9 ?

9000005.0)!1(

3,

!103628800600000362880!9

3628800!10

362880)1)(2)(3)(4)(5)(6)(7)(8(9!9

,

600000)!1(

000005.0)!1(

3

nisn

satisfyingnsmallesttheThus

haveWe

n

n

Let

Example (2)Approximate sin3○ to five decimal places

753

12

1

!7

10

!5

10

!3

100

)!12()1(sin

sin

,

xxxx

n

xx

isxforpolynomialMaclaurinThe

haveWe

n

n

n

1

)1(

)1(

11

753

0

12

)60()!1(

1)

60(

1,1cos,sin

cossin)(

)}50,1()({

)60(

)!1(0

60)!1()

60(

!7

10

!5

10

!3

100

)!12()1()(

sin

nn

n

n

nn

n

kn

k

kk

n

nR

choosewexxand

xorxtoequaleitherisxfBecause

xxfsettheforboundupperanyiswhere

nnR

saystheoremThe

xxxx

xk

xxp

isxforpolynomialMaclaurinThe

5234.0

)60(

31

10

6003sin

:53sin

3

000005.0)60()!1(

1

000005.0

3

1

isplacesdecimaltoaccurateofionapproximatan

nisinequalityabovethesatisfyingnsmallestThe

n

Let

toleratederrorThe

n

Approximating the sum of a convergent alternating series

Theorem

Let T be the sum of an alternating series satisfying the main convergence test for an alternating series. Then:

1. T lies between any two successive partial sums of the series

2. If T is approximated by a partial sum Tn, then:

a. |T - Tn | ≤ an+1

b. The sign of the error is the same as that of the coefficient of an+1

Using power series to approximate definite integrals

Example

Give an estimation of the integral on the right accurate to 3 decimal places

dxe x 1

0

2

A power series for the given integral

0

1

0

1

0

28642

1

0

!)12(

)1(

!4)9(

1

!3)7(

1

!2)5(

1

3

11

9753

[

]!

)1(

!4!3!21[

]!4)9(!3)7(!2)5(3

2

n

n

nn

x

nn

dxn

xxxxx

dxe

xxxxx

Approximating the integral

747.0

!5)9(

1

!4)9(

1

!3)7(

1

!2)5(

1

3

11

5,

0005.0)!1)(32(

1

,0005.0

)!1)(32(

1

1

0

1

1

0

2

2

dxe

nisthatsatisfyingnsmallestthe

nn

letweaccuracyrequiredthegetTo

nnaTdxe

seriestheofTsumpartialanyFor

x

nnx

n

Homework

001.)4.1ln()4(

.

max),1.0()3(

001.0)2

1sin().2(

2sin)1(

410

withineApproximat

ionapproximatsuchoferror

imumthecalculatethenandpbyeeApproximat

withineApproximat

placesdecimalfourtoeApproximat

1

)1(

1

53

0349.0)!1(

1)0349.0(

,

?)(1

})1,1(:)({

00349.0)!1(

)0349.0(

!5

10

!3

100sin

0349.02360

22

2sin

)1(

n

n

n

n

n

nR

Thus

Whyexamplefor

xxf

settheforboundupperenoughsmallanyiswhere

nR

saystheoremThe

xxxx

toscorrespond

xxxp

thatNotice

p

Thus

nneedweThen

Rbecause

nthatsuchnfindWe

thanlessbeRremainderthethatneedWe

n

n

00)(

0349.0

0)0349.0(0)0349.0(

2

00004.00349.0!3

1)0349.0(

00005.00349.0)!1(

1,

00005.0)0349.0(

2

2

3

2

1

1

1

)1(

1

53

2)!1(

1

2

1

)!1(

1)2

1(

,

?1

})1,0(:)({

02

1

)!1()2

1(

!5

1

!3

1sin

2

1sin

)2(

n

n

n

n

n

n

nnR

Thus

Whyexamplefor

xxf

settheforboundupperenoughsmallaiswhere

nR

saystheoremThe

xxxx

001.03840

1

2!5

1

2)!14(

1

?4

:48

23

48

1

2

10)

2

1(!3

10)

2

1(0

2

1sin

0)(!3

100)(

2

1sin)

2

1(

4

001.02)!1(

1,

514

3

34

4

1

npickedweWhy

Question

xxxp

thatNotice

accuracygiventhewithinofionapproximattheispThus

n

nthatsuchnneedWe

n

sumspartialucativetwoanybetweenliessumThe

thatsaysseriesgalternatinofsumtheoftheoremThe

seriesgalternatinofeconvergencthefortestmainthesatisfiesseriesThe

n

xxxxxx

n

nn

cos

:

)4.0(1

)1(

)4.0(5

1)4.0(

4

1)4.0(

3

1)4.0(

2

14.0

)4.01ln(5

1

4

1

3

1

2

1)1ln(

10

14ln

)4(

1

5432

5432

34.010

14ln

:,

341.010

14ln335.0

)4.0(3

1)4.0(

2

14.0

10

14ln)4.0(

4

1)4.0(

3

1)4.0(

2

14.0

0064.0)4.0(4

1:01.0

32432

4

exampleforisaccuracyrequiredthesatisfyingionapproximatanThus

isthanlesstermfirstThe