ALTERNATING SERIES

series with positive terms6

1

5

1

4

1

3

1

2

11

7

1

6

1

5

1

4

1

3

1

2

11

series with some positive and some negative terms

6

1

5

1

4

1

3

1

2

11 alternating series

1

1)1(

n

n

n

evenn

oddnn

1

11)1(

n-th term of the series

nn una 1)1(

nu are positive

ALTERNATING SERIES

alternating series

6

1

5

1

4

1

3

1

2

11

)1(

1

1

n

n

n

6

1

8

1

4

1

2

1)

2

1(

1

n

n

alternating harmonic series

alternating geomtric series

alternating p-series

ppppn

p

n

n 5

1

4

1

3

1

2

11

)1(

1

1

THEOREM: (THE ALTERNATING SERIES TEST)

nn uu 1

ALTERNATING SERIES

0lim n

nu

)1

)2

)3

0nu

1

1)1(n

nn u

alternating

decreasing

lim = 0

1

1)1(n

nn u

convg

Remark:The convergence tests that we have looked at so far apply only to series with positive terms. In this section and the next we learn how to deal with series whose terms are not necessarily positive. Of particular importance are alternating series, whose terms alternate in sign.

Determine whether the series converges or diverges.

Example:

1

1)1(

n

n

n

ALTERNATING SERIES

Determine whether the series converges or diverges.

Example:

13

21

1)1(

n

n

n

n

THEOREM: (THE ALTERNATING SERIES TEST)

nn uu 1

0lim n

nu

)1

)2

)3

0nu

1

1)1(n

nn u

alternating

decreasing

lim = 0

1

1)1(n

nn u

convg

ALTERNATING SERIES

Determine whether the series converges or diverges.

Example:

1 14

3)1(

n

n

n

n

THEOREM: (THE ALTERNATING SERIES TEST)

nn uu 1

0lim n

nu

)1

)2

)3

0nu

1

1)1(n

nn u

alternating

decreasing

lim = 0

1

1)1(n

nn u

convg

THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM)

ALTERNATING SERIES

1

1)1(n

nn u satisfies the three

conditions

approximates the sum L of the series with an error whose absolute value is less than the absolute value of the first unused term

1

1)1(i

ii uL

n

ii

in uS

1

1)1(

nS

1nu

the sum L lies between any two successive partial sums andnS

1nS

the remainder, has the same sign as the first unused term.

nn SLR

THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM)

1

1)1(n

nn u

satisfies the three conditions

1 nn uSL

1 nn SLS

)(

)(

unused1st sign

SLsign n

nn SLS 1

OR

Example: 666.02

1)1(

11

1

nn

n

32

15 SL

negativeSLsign )( 5

6875.05665625.0 SLS

0.687516

1

8

1

4

1

2

115 S

0.6562532

156 SS

ALTERNATING SERIES

Find the sum of the series correct to three decimal places.

Example:

0

1

!

)1(

n

n

n

1

1)1(i

ii uL

n

ii

in uS

1

1)1(

THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM)

1

1)1(n

nn u

satisfies the three conditions

1 nn uSL

1 nn SLS

)(

)(

unused1st sign

SLsign n

nn SLS 1

OR

ALTERNATING SERIES

Find the sum of the series correct to three decimal places.

Example:

0

1

!

)1(

n

n

n

ALTERNATING SERIES

The rule that the error is smaller than the first unused term is, in general, valid only for alternating series that satisfy the conditions of the Alternating Series Estimation Theorem. The rule does not apply to other types of series.

REMARK:

1

1)1(i

ii uL

n

ii

in uS

1

1)1(

THEOREM: (THE ALTERNATING SERIES ESTIMATION THEOREM)

1

1)1(n

nn u

satisfies the three conditions

1 nn uSL

1 nn SLS

)(

)(

unused1st sign

SLsign n

nn SLS 1

OR

ALTERNATING SERIES

TERM-102

TERM-101

ALTERNATING SERIES

TERM-092

ALTERNATING SERIES

6

1

5

1

4

1

3

1

2

11

1nna

22222

1 6

1

5

1

4

1

3

1

2

11

nna

6

1

5

1

4

1

3

1

2

11

)1(

1

1

n

n

n

6

1

5

1

4

1

3

1

2

11

1nna

22222

1 6

1

5

1

4

1

3

1

2

11

nna

6

1

5

1

4

1

3

1

2

11

1

1n n

Is called Absolutely convergent

DEF:

1nna

1nna

convergent

IF

converges absolutely

Test the series for absolute convergence.

Example:

12

1)1(

n

n

n

Alternating Series, Absolute and Conditional Convergence

Is called Absolutely convergent

DEF:

1nna

1nna

convergent

IF

converges absolutely

Test the series for absolute convergence.

Example:

12

)sin(

n n

n

Is called conditionally convergent

DEF:

1nna Test the series for absolute

convergence.

Example:

1

1)1(

n

n

n

if it is convergent but not absolutely convergent.

REM:

1nna

1nna

convg divg

Alternating Series, Absolute and Conditional Convergence

Is called Absolutely convergent

DEF:

1nna

1nna

convergent

IF

converges absolutely

Test the series for absolute convergence.

Example:

Is called conditionally convergent

DEF:

1nna

if it is convergent but not absolutely convergent.

REM:

1nna

1nna

convg divg

Alternating Series, Absolute and Conditional Convergence

Absolutely convergentTHM:

1nna convergent

1nna

Determine whether the series converges or diverges.

Example:

12

cos

n n

n

Alternating Series, Absolute and Conditional Convergence

convgTHM:

1nna convg

1nna

Absolutely convergent

conditionally convergent

1nna

1nna

convergent

divergent

1nna

1nna

1nna

1nna

Alternating Series, Absolute and Conditional Convergence

Is called Absolutely convergent

DEF:

1nna

1nna

convergent

IF

converges absolutely

Choose one: absolutely convergent or conditionally convergent

Example:

Is called conditionally convergent

DEF:

1nna

if it is convergent but not absolutely convergent.

REM:

1nna

1nna

convg divg

Alternating Series, Absolute and Conditional Convergence

REARRANGEMENTS

1

)1(n

nDivergent

111111111

1)11()11()11()11(1

0)11()11()11()11(

If we rearrange the order of the terms in a finite sum, then of course the value of the sum remains unchanged.

But this is not always the case for an infinite series.

By a rearrangement of an infinite series we mean a series obtained by simply changing the order of the terms.

Alternating Series, Absolute and Conditional Convergence

REARRANGEMENTS

1

)1(n

nDivergent

111111111

1)11()11()11()11(1

0)11()11()11()11(

1

1)1(

n

n

nconvergent

7

1

6

1

5

1

4

1

3

1

2

11

2ln17

1

6

1

5

1

4

1

3

1

2

1

2ln12

1

7

1

6

1

5

1

4

1

3

1

2

1

See page 719

Alternating Series, Absolute and Conditional Convergence

REARRANGEMENTS

Absolutely convergent

REMARK:

1nna

with sum s

any rearrangement has the same sum s

Conditionallyconvergent

Riemann proved that

1nna

r is any real number

there is a rearrangement that has a sum equal to r.

Alternating Series, Absolute and Conditional Convergence

SUMMARY OF TESTS

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

5) Alternating 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

SUMMARY OF TESTS

1

)1(

np

n

n

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-101

ALTERNATING SERIES