1. sequences and series

8/12/2019 1. Sequences and Series

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30: Sequences and Series

Christine Crisp

Teach A Level Maths

Vol. 1: AS Core Modules


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Sequences and Series

Module C1

AQAEdexcel

OCR

MEI/OCR

Module C2

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permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"
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Examples of Sequences

e.g. 1 ...,8,6,4,2

e.g. 2 ...,4

1,

3

1,

2

1,1

e.g. 3 ...,64,16,4,1 A sequence is an ordered list of numbers

The 3 dots are used to show that a sequence continues


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Recurrence Relations

...,9,7,5,3Can you predict the next term of the sequence

?

Suppose the formula continues by adding 2 toeach term.

The formula that generates the sequence is then

21 nn uu

223 uu

where and are terms of the sequencen

u 1nu

is the 1st

term, so1u 31u5232 u7253 u

etc.

1n 212 uu2n

11


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nn uu 41

e.g. 1 Give the 1stterm and write down a

recurrence relation for the sequence...,64,16,4,1

1stterm: 11uSolution:

Other letters may be used instead of uand n, sothe formula could, for example, be given as

kk aa 41

Recurremce relation:

A formula such as is called arecurrence relation

21 nn uu


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e.g. 2 Write down the 2nd

, 3rd

and 4th

terms ofthe sequence given by 32,5 11 ii uuu

1iSolution: 32 12 uu73)5(22 u

2i 32 23 uu113)7(23 u

3i 32 34 uu193)11(24 u

The sequence is ...,19,11,7,5


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Properties of sequences

Convergent sequences approach a certain value

e.g. approaches 2...1,1,1,1,11615

87

43

21

n

nu


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e.g. approaches 0...,,,,,1161

81

41

21

This convergent sequence also oscillates


n

nu


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e.g. ...,16,8,4,2,1

This divergent sequence also oscillates

Divergent sequences do not converge

n

nu


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e.g. ...,3,2,1,3,2,1,3,2,1

This divergent sequence is also periodic


n

nu
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Convergent ValuesIt is not always easy to see what value a sequence

converges to. e.g.

n

n

n

u

uuu

310,1 11

...,11

103

,7

11

,7,1 The sequence isTo find the value that the sequence converges towe use the fact that eventually ( at infinity! ) the

(n + 1

)th

term equals then

th

term.Let . Then,uuu

nn 1

u

uu

310 0103

2 uu0)2)(5( uu 25 uu since

uu 3102 Multiply by u :


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Sequences and SeriesExercises1. Write out the first 5terms of the following

sequences and describe the sequence using the

words convergent, divergent, oscillating,periodic as appropriate

(b)n

nu

uu 12 11 and

2. What value does the sequence given by ,u 21

34 11 nn uuu and(a)

nn uuu

2

111 16 and(c)

Ans: 8,5,2,1,4 Divergent

Ans: 2,,2,,221

21 Divergent Periodic

Ans: 1,2,4,8,16 Convergent Oscillatinguuu

nn 1Let

370330 uuu 730

uto?converge3301 nn uu
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General Term of a Sequence

Some sequences can also be defined by giving ageneral term. This general term is usually called then

thterm.

n2

n

1

The general term can easily be checked by substitutingn= 1, n= 2, etc.

e.g. 1 n

u...,8,6,4,2

e.g. 2 nu...,4

1,

3

1,

2

1,1

e.g. 3 nu...,64,16,4,1 1)4( n
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Exercises

Write out the first 5terms of the followingsequences1.

(b) nnu )2(nun 41(a)

22nun(c)n

nu )1((d)

19,15,11,7,3 32,16,8,4,2

50,32,18,8,2

1,1,1,1,1 Give the general term of each of the following sequences2.

...,7,5,3,1(a) 12nun...,243,81,27,9,3 (c)

(b) ...,25,16,9,4,1

(d) ...,5,5,5,5,5 5)1( 1 nnu

2nun

n

nu )3(
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Sequences and SeriesSeries

When the terms of a sequence are added, we get aseries

...,25,16,9,4,1The sequence

gives the series ...2516941 Sigma Notation for a Series

A series can be described using the general term

100...2516941 e.g.

10

1

2n

can be written

is the Greek capital letter S, used for Sum1stvalue of n

last value of n


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16...8642 (a) 81

2n

1003...2793 (b)

2. Write the following using sigma notation

Exercises

1. Write out the first 3 terms and the last term ofthe series given below in sigma notation

201

12n(a) 1

1024...842 (b) 101

2 n

3n= 1n= 2

39...5

1001

3 n

n= 20


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The following slides contain repeats ofinformation on earlier slides, shown withoutcolour, so that they can be printed and

photocopied.

For most purposes the slides can be printedas Handouts with up to 6slides per sheet.


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nn uu 41

e.g. 1 Give the 1stterm and write down a

recurrence relation for the sequence...,64,16,4,1

1stterm: 11uSolution:

Other letters may be used instead of uand n, sothe formula could, for example, be given as

kk aa 41

Recurremce relation:

A formula such as is called arecurrence relation

21 nn uu


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e.g. Write down the 2nd

, 3rd

and 4th

terms ofthe sequence given by 32,5 11 ii uuu1iSolution: 32 12 uu

73)5(22 u2i 32 23 uu

113)7(23 u3i 32 34 uu

193)11(24 uThe sequence is ...,19,11,7,5


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e.g. approaches 2...1,1,1,1,11615

87

43

21

n

nu


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e.g. approaches 0...,,,,,1161

81

41

21

This convergent sequence also oscillates


n

nu


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e.g. ...,16,8,4,2,1

This divergent sequence also oscillates


n

nu


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e.g. ...,3,2,1,3,2,1,3,2,1

This divergent sequence is also periodic


n

nu


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Convergent ValuesIt is not always easy to see what value a sequence

converges to. e.g.

n

n

n

u

uuu

310,1 11

...,11

103

,7

11

,7,1 The sequence isTo find the value that the sequence converges towe use the fact that eventually ( at infinity! ) the( n + 1) thterm equals the n thterm.

Let . Then,uuunn

1u

uu

310 0103

2 uu0)2)(5( uu 25 uu since

uu 3102 Multiply by u :


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General Term of a Sequence

Some sequences can also be defined by giving ageneral term. This general term is usually called then

thterm.

n2

n

1

The general term can easily be checked by substitutingn= 1, n= 2, etc.

e.g. 1 n

u...,8,6,4,2

e.g. 2 nu...,4

1,

3

1,

2

1,1

e.g. 3 nu...,64,16,4,1 1)4( n


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Sequences and SeriesSeries

When the terms of a sequence are added, we get aseries

...,25,16,9,4,1The sequence

gives the series ...2516941 Sigma Notation for a Series

A series can be described using the general term

100...2516941 e.g.

10

1

2n

can be written

is the Greek capital letter S, used for Sum1stvalue of n

last value of n

1. sequences and series

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