Final Exam
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Transcript of Final Exam
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