SERIES

A sequence is a list of numbers written in a definite order:

, , , ,}{ 211 nii aaaa

DEF:

Is called a series

21 iaaaDEF:

1iia

Example:

1 2

1

ii

, , , , ,}2

1{

2

1

8

1

4

1

2

1 1 iii

16

1

8

1

4

1

2

1 1

SERIES

Is called a series

21 iaaaDEF:

1iia

Example:

1 2

1

ii

16

1

8

1

4

1

2

1 1

n-th termits sum

convergentDEF:

1iiaIf the sum of the series

is finite number not infinity

SERIES

nth-partial sums :

DEF: Example:

1 2

1

nn

n

n

iin aaaas

211

Given a seris

1iia

2

11 s

4

1

2

12 s

8

1

4

1

2

13 s

16

1

8

1

4

1

2

14 s

32

1

16

1

8

1

4

1

2

15 s

5.01 s

75.02 s

875.03 s

9375.04 s

96875.05 s

the sequence of partial sums. :

DEF:

,, 211 sss nn

Given a seris

1iia

} ,96875.0

,9375.0 ,875.0 ,75.0 ,5.0 {

ns

SERIES

Given a series

1iia

We define

Sequence of partial sums

1nns

Given a series Sequence of partial sums

1 2

1

nn

} ,96875.0

,9375.0 ,875.0 ,75.0 ,5.0 {

ns

SERIES

Given a series

1iia

We define

Sequence of partial sums

1nns

DEF: 1nnsIf convergent

1iia convergent

nn

ii sa

lim1

1nnsIf divergent

1iia divergent

SERIES

Example:

1 2

1

nn

2

11 s

4

1

2

12 s

8

1

4

1

2

13 s

16

1

8

1

4

1

2

14 s

32

1

16

1

8

1

4

1

2

15 s

5.01 s

75.02 s

875.03 s

9375.04 s

96875.05 s

} ,96875.0

,9375.0 ,875.0 ,75.0 ,5.0 {

ns

SERIES

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

Geometric Series:

1

1

n

nar

first term

Common ratio

ar

Example

1

13

1)(4n

n

2arara

nn raa 1

27

4

9

4

3

44Example:

1 2

1

nn

ar

Is it geometric?

Example

5

8

5

4

5

2

5

1

10

1

Is it geometric?

1

)1(

np

n

n

SERIES

Geometric Series:

1

1

n

nar

Example

1

13

1)(4n

n

2arara

27

4

9

4

3

44

Example:

1 2

1

nn

Example

5

8

5

4

5

2

5

1

10

1

Is it geometric?

Geometric Series:

1

1

n

nar

1

1 1

rdivg

rconvgr

a

SERIES

Final-111

SERIES

Final-102

SERIES

Geometric Series:

1

1

n

nar 2arara

Geometric Series:

1

1

n

nar

1

1 1

rdivg

rconvgr

a

prove: nn rss

SERIES

Geometric Series:

1

1

n

nar 2arara

Geometric Series:

1

1

n

nar

1

1 1

rdivg

rconvgr

a

SERIES

Final-102

SERIES

Geometric Series:

1

1

n

nar 2arara

Geometric Series:

1

1

n

nar

1

1 1

rdivg

rconvgr

a

SERIES

Telescoping Series:

11)(

nnn bb

11 nn bbs

111

1 lim)(

n

nn

nn bbbbs

Telescoping Series:

Convergent

} { nb

11)(

nnn bb

Convergent

1

)1

11(n nn

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

1

)1(

np

n

n

Example:

Remark: b1 means the first term ( n starts from what integer)

SERIES

Final-111

11 nn bbs

111

1 lim)(

n

nn

nn bbbbs

Telescoping Series:

Convergent

} { nb

11)(

nnn bb

Convergent

SERIES

Telescoping Series:

11)(

nnn bb

11 nn bbs

111

1 lim)(

n

nn

nn bbbbs

Telescoping Series:

Convergent

} { nb

11)(

nnn bb

Convergent

Notice that the terms cancel in pairs. This is an example of a telescoping sum: Because of all the cancellations, the sum collapses (like a pirate’s collapsing telescope) into just two terms.

SERIES

Final-101

Final-112

SERIES

Final-101

SERIES

Convergent

1iia

0lim n

na

THEOREM:

Divergent

1iia 0lim

nna

THEOREM:THE TEST FOR DIVERGENCE

DNE lim nna

or

SERIES

Divergent

1iia 0lim

nna

THEOREM:THE TEST FOR DIVERGENCE

DNE lim nna

or

SERIES

Divergent

1iia 0lim

nna

THEOREM:THE TEST FOR DIVERGENCE

DNE lim nna

or

Convergent

1iia 0lim

nnaTHEOREM:

REMARK(1):

The converse of Theorem is not true in general. If we cannot conclude that is convergent.

0lim n

na

1iia

If we find that 0lim n

na

we know nothing about the convergence or divergence

REMARK(2):the set of all series

0lim na

Convergent

0lim na

na

SERIES

series

1iia

Sequence

1nns

Convergent

1iia

0lim n

naTHEOREM:

Seq.

1nna

convg

1iia

convg

1nns

1nna

0convg

REMARK(2):

REMARK(3):

SERIES

REMARK

All these items are true if these two series are convergent

Example

111111

111111

SERIES

Final-082

SERIES

Final-081

SERIES

Final-092

SERIES

Final-121

SERIES

Final-121

SERIES

Final-103

SERIES

SERIES

REMARK(4):

A finite number of terms doesn’t affect the divergence of a series.

12

2

45n n

nExample

REMARK(5):

A finite number of terms doesn’t affect the convergence of a series.

REMARK(6):

A finite number of terms doesn’t affect the convergence of a series but it affect the sum.

n

n

)(51

2

1

Example

n

n

)(53

2

1

Adding or Deleting Terms

102

2

45n n

n

5

4

5

SERIES

1 2

1

nn

Example

Reindexing

32

1

16

1

8

1

4

1

2

1

We can write this geometric series

5nm

5mn

652

1

mm

3nm

3mn

232

1

mm

SERIES

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

1

)1(

np

n

n

SERIES summary

GeometricTelescopingGeneral

When convg

sum

nth partial sum11 nn bbs

11 lim nnbb

convgbn } { 11r

r

a

1

r

ras

n

n

1

1

convgsn} {

nns

lim

nn aas 1

1 nnn ssa

Divergent

1iia 0lim

nna

THEOREM:THE TEST FOR DIVERGENCE

DNE lim nna

or

convg

1iia

convg

1nns

1nna

0convg

SERIES

Geometric Series:

1

1

n

nar 2arara

Geometric Series:

1

1

n

nar

1

1 1

rdivg

rconvgr

a

Example

9595959595.25

Write as a ratio of integers

333333.0

SERIES

Final-101

SERIES

Final-112