30: sequences and series © christine crisp “teach a level maths” vol. 1: as core modules
TRANSCRIPT
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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Sequences and Series
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
Sequences and SeriesRecurrence
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
Sequences and SeriesRecurrence
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 and SeriesProperties of
sequences
e.g. approaches 0...,,,,,1161
81
41
21
This convergent sequence also
oscillates
Convergent sequences approach a
certain value
n
nu
Sequences and SeriesProperties of
sequences
e.g. ...,10,8,6,4,2
Divergent sequences do not
converge
n
nu
Sequences and SeriesProperties of
sequences
e.g. ...,16,8,4,2,1
This divergent sequence also
oscillates
Divergent sequences do not
converge
n
nu
Sequences and SeriesProperties of
sequences
e.g
.
...,3,2,1,3,2,1,3,2,1
This divergent sequence is also
periodic
Divergent sequences do not
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 :
Sequences and Series
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
Sequences and Series
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
Sequences and Series
Sequences and Series
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.
Sequences and SeriesRecurrence
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
Sequences and SeriesRecurrence
Relationse.g. 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 and SeriesProperties of
sequences
e.g. approaches 0...,,,,,1161
81
41
21
This convergent sequence also
oscillates
Convergent sequences approach a
certain value
n
nu
Sequences and SeriesProperties of
sequences
e.g. ...,16,8,4,2,1
This divergent sequence also
oscillates
Divergent sequences do not
converge
n
nu
Sequences and SeriesProperties of
sequences
e.g
.
...,3,2,1,3,2,1,3,2,1
This divergent sequence is also
periodic
Divergent sequences do not
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 :
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 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