30: sequences and series © christine crisp “teach a level maths” vol. 1: as core modules

30: Sequences and 30: Sequences and SeriesSeries

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Sequences and Series

Module C1

AQAEdexcel

OCR

MEI/OCR

Module C2

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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

Sequences and SeriesRecurrence

Relations

...,9,7,5,3

Can you predict the next term of the sequence

?Suppose the formula continues by adding 2 to each term.The formula that generates the sequence is then

21 nn uu

223 uu

where and are terms of the sequencenu 1nu

is the 1st term, so1u 31 u5232 u

7253 u

etc.

1n 212 uu

2n

11


Relations

nn uu 41

e.g. 1 Give the 1st term and write down a

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

1st term: 11 uSolution:

Other letters may be used instead of u and n, so the formula could, for example, be given as

kk aa 41

Recurremce relation:

A formula such as is called a

recurrence relation

21 nn uu


Relationse.g. 2 Write down the 2nd, 3rd and 4th terms of

the sequence given by 32,5 11 ii uuu

1iSolution: 32 12 uu

73)5(22 u

2i 32 23 uu

113)7(23 u

3i 32 34 uu

193)11(24 uThe sequence

is ...,19,11,7,5

Sequences and SeriesProperties of

sequencesConvergent sequences approach a

certain value

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

87

43

21

n

nu


sequences

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

81

41

21

This convergent sequence also

oscillates

Convergent sequences approach a

certain value

n

nu


sequences

e.g. ...,10,8,6,4,2

Divergent sequences do not

converge

n

nu


sequences

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

This divergent sequence also

oscillates


converge

n

nu


sequences

e.g

.

...,3,2,1,3,2,1,3,2,1

This divergent sequence is also

periodic


converge

n

nu

Sequences and SeriesConvergent

ValuesIt is not always easy to see what value a

sequence converges to. e.g.

n

nn u

uuu

310,1 11

...,11

103,

7

11,7,1

The sequence

isTo find the value that the sequence converges to we use the fact that eventually ( at infinity! ) the ( n + 1 ) th term equals the n th term.

Let . Then, uuu nn 1 u

uu

310

01032 uu

0)2)(5( uu 25 uu since

uu 3102 Multiply by u :


Exercises1. Write out the first 5 terms of the following sequences and describe the sequence using the words convergent, divergent, oscillating, periodic as appropriate

(b) n

n uuu

12 11 and

2. What value does the sequence given by

,u 21

34 11 nn uuu and (a)

nn u uu 21

11 16 and (c)

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

Ans:

2,,2,,221

21 Divergent

Periodic

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

uuu nn 1Let

370330 uuu7

30 u

to? converge 3301 nn uu

Sequences and SeriesGeneral Term of a

SequenceSome sequences can also be defined by giving a general term. This general term is usually called the nth term.

n2

n

1

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

e.g. 1

nu...,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

Sequences and SeriesExercise

sWrite out the first 5 terms of the following sequences

1.

(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 sequences

2.

...,7,5,3,1(a

) 12 nun

...,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(

Sequences and SeriesSeries

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

...,25,16,9,4,1The sequencegives the series

...2516941

Sigma Notation for a SeriesA series can be described using the general

term100...2516941 e.g.

10

1

2ncan be written

is the Greek capital letter S, used for Sum

1st value of n

last value of n


16...8642 (a)

8

1

2n

1003...2793 (b)

2. Write the following using sigma notation

Exercises1. Write out the first 3 terms and the last term of the series given below in sigma notation

20

1

12n(a

) 1

1024...842 (b)

10

1

2 n

3n = 1n = 2

39...5

100

1

3 n

n = 20


The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.