Final Exam

Term121 Term112Improper Integral

and Ch10 16 16Others 12 12

Term121 Term112Others

(Techniques of Integrations)

8 8

Others-Others 4 4

Remark: (24)

1) Chapter10

2) Improper Integral

3) Techniques of Integration

SUMMARY OF Power Series

1) How to find Maclaurin series

)1ln(,)1(,tan,,sin,cos,1

1 1 xxxexxx

kx

Memorize

subsitute differentiate2xxe

product

2

3

)3( x

x

xex cos

formula

n

n

n

xn

f

0

)(

!

)0(

integrate

)(tan 1 x

2) Find: radius and inteval of convergence

0

)()(n

nn axcxf

1)(

)(lim

11

L

axc

axcn

n

nn

nRax

Study the endpoints for interval of convergence

3) Binomial Series

2

!2

)1(1)1( x

kkkxx k

5) Applications of Power Series

Find the sum Find integral

0 12

)1(

n

n

n 1

0

2 )sin( dxx

4) Taylor Series

0

)(

)(!

)()(

n

nn

axn

afxf

1)1(

)()!1(

)()(

nn

n axn

cfxR

1 5nn

n

MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620

TERM-121

TAYLOR AND MACLAURIN

TERM-092

TAYLOR AND MACLAURIN

TERM-081

TERM-121

TERM-121

Special Series:

1) Geometric Series

2) Harmonic Series

3) Telescoping Series

4) p-series

5) Alternating p-series

1

1

n

nar

1

1

nn

11)(

nnn bb

1

1

npn

SUMMARY OF TESTS

1

)1(

np

n

n

P Series:

1

11

1 pdivg

pconvg

nnp

10 .

1 .)1(

1

1

pconvgCond

pconvgAbs

nnp

n

Alter. P Series:

Series Tests

1) Test for Divergence

2) Integral Test

3) Comparison Test

4) Limit Comparison Test

5) Ratio Test

6) Root Test

7) Alternating Series Test

0lim n

na

1)( dxxf

nn ab

n

n

n b

ac

lim

n

n

n a

a 1lim

nn

na

lim

0lim,,decalt

SUMMARY OF TESTS Integral Test:

1nna

1

)( dxxf

Both convg or divg

Comparison Test:

Smaller divg bigger divg

nn ab

Bigger convg Smaller convg

Limit Comparison Test (1):Both convg or divg c0

Limit Comparison Test (2):

0c nn ab

c nn ba

Ratio & Root fail

divg

conv

1

1

1

5-types

1) Determine whether convg or divg 2) Find the sum s

3) Estimate the sum s

4) How many terms are needed

within error

5) Partial sums

SUMMARY OF TESTS

TERM-121

TERM-121

TERM-121

10 .

1 .)1(

1

1

pconvgCond

pconvgAbs

nnp

n

Alter. P Series:

! factorial n

Ratio Test

nnb )(n ofpower

Root Test

MIX

1

)(n

nf

)(xf easy to integrate

Integral Test

Faster

nn

!n

,3 ,2 nn

Absolutely convergent

conditionally convergent

1nna

1nna

1nna

1nna

convergent divergent

convergent convergent

Alternating Series, Absolute and Conditional Convergence

Absolutely convergentTHM:

1nna convergent

1nna

convgTHM:

1nna convg

1nna

1

2

)!12(

)!()1(

n

n

n

n

11

2

5

35)1(

nnn

nn

n

TERM-121

TERM-091

Recursively Defined Terms

SERIES

series

1iia

Sequence

1nns

Convergent

1iia

0lim n

naTHEOREM:

Seq.

1nna

convg

1iia

convg

1nns

1nna

0convg

REMARK(2):

REMARK(3):

TERM-121