Final Exam
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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
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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
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MEMORIZE: ** Students are required to know the series listed in Table 10.1, P. 620
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TERM-121
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TAYLOR AND MACLAURIN
TERM-092
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TAYLOR AND MACLAURIN
TERM-081
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TERM-121
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TERM-121
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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:
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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
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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
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TERM-121
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TERM-121
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TERM-121
10 .
1 .)1(
1
1
pconvgCond
pconvgAbs
nnp
n
Alter. P Series:
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! 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
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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
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1
2
)!12(
)!()1(
n
n
n
n
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11
2
5
35)1(
nnn
nn
n
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TERM-121
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TERM-091
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Recursively Defined Terms
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SERIES
series
1iia
Sequence
1nns
Convergent
1iia
0lim n
naTHEOREM:
Seq.
1nna
convg
1iia
convg
1nns
1nna
0convg
REMARK(2):
REMARK(3):
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TERM-121