Sec 11.7: Strategy for Testing Series Series Tests 1)Test for Divergence 2) Integral Test 3)...
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Sec 11.7: Strategy for Testing 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 Special Series: 1) Geometric Series 2) Harmonic Series 3) Telescoping Series 4) p-series 1 1 n n ar 1 1 n n 1 1 ) ( n n n b b 1 1 n p n 0 lim n n a 1 ) ( dx x f n n a b n n n b a c lim n n n a a L 1 lim n n n a L lim 0 lim , , dec alt
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Transcript of Sec 11.7: Strategy for Testing Series Series Tests 1)Test for Divergence 2) Integral Test 3)...
Sec 11.7: Strategy for Testing 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
Special Series:
1) Geometric Series
2) Harmonic Series
3) Telescoping Series
4) p-series
1
1
n
nar
1
1
nn
11)(
nnn bb
1
1
npn
0lim n
na
1)( dxxf
nn ab
n
n
n b
ac
lim
n
n
n a
aL 1lim
nn
naL
lim
0lim,,decalt
Sec 11.7: Strategy for Testing Series
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
1nna
STRATEGY FOR TESTING SERIES p-series
1
1
npn
geometric
1
1
n
nar
Similar to geom or p-series
1 3
1
npn
1 23
1
nn
Use Comparison or Limit Comparison
0lim n
na
Test for DivergenceAlternating
1
)1(n
nnb
! factorial n
Ratio Test
nnb )(n ofpower
Root Test
1
)(n
nf
)(xf easy to integrate
Integral Test
telescoping
11)(
nnn bb
Sec 11.7: Strategy for Testing Series
TERM-091
Sec 11.7: Strategy for Testing Series
TERM-082
Sec 11.7: Strategy for Testing Series
TERM-101
Sec 11.7: Strategy for Testing Series
TERM-102
Sec 11.7: Strategy for Testing Series
TERM-102
Sec 11.7: Strategy for Testing Series
Remark:All terms are not positive
TERM-091
Sec 11.7: Strategy for Testing Series
TERM-082
Sec 11.7: Strategy for Testing Series
TERM-091
Sec 11.7: Strategy for Testing Series
TERM-091
Sec 11.7: Strategy for Testing Series
TERM-082
Sec 11.7: Strategy for Testing Series
TERM-102
Sec 11.7: Strategy for Testing Series
Remark:All terms are not positive
TERM-082
Sec 11.7: Strategy for Testing Series
TERM-102
Sec 11.7: Strategy for Testing Series
TERM-102
Sec 11.7: Strategy for Testing Series
TERM-092
Sec 11.7: Strategy for Testing Series
TERM-082
Sec 11.7: Strategy for Testing Series
Sec 11.7: Strategy for Testing Series
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
Geometric Series:
1
1
n
nar
1
1 1
rdivg
rconvgr
a
2) Find the sum s
11 nn bbs
111
1 lim)(
n
nn
nn bbbbs
Telescoping Series:
Convergent
} { nb
11)(
nnn bb
Convergent
nth-partial sums :
DEF:
n
n
iin aaaas
211
Given a seris
1iia
nnn ssa 1
nth-partial sums :
DEF:
n
n
iin aaaas
211
Given a seris
1iia
nnn ssa 1
3) Estimate the sum s
TERM-101
TERM-091
Sec 11.7: Strategy for Testing Series
Final-092
4) How many terms are needed within error
REMAINDER ESTIMATE FOR THE INTEGRAL TEST
nnndxxfRdxxf )()(
1
THEOREM: (ALTERNATING SERIES ESTIMATION THEOREM)
1
1)1(n
nn b
nn bb 1
0lim n
nb
)1)2
)3
0nb 1 nn bR
REMAINDER ESTIMATE FOR THE INTEGRAL TEST
nnndxxfRdxxf )()(
1
THEOREM: (ALTERNATING SERIES ESTIMATION THEOREM)
1
1)1(n
nn b
nn bb 1
0lim n
nb
)1)2
)3
0nb 1 nn bRTERM-102
TERM-101
n S_n |R_n| b_(n+1)
7 -0.11393129032210 0.0011582566 0.0019569
8 -0.11588823748453 0.0007986905 0.0013736
10 -0.11551561211190 0.0004260651 0.0007518
12 -0.11534277120825 0.0002532242 0.0004553
999 -0.11508905392165
1000 -0.11508905492165
TERM-092
Sec 11.7: Strategy for Testing Series
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
5) Partial sums
Geometric Series:
11 nn bbs
Telescoping Series:
DEF:
n
n
iin aaaas
211
Given a seris
1iia
nnn ssa 1
nn raa 1
r
ras
n
n
1
)1(
