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(