This document discusses sequences and series. It defines a sequence as an ordered list of numbers and uses examples to show common types of sequences like convergent, divergent, oscillating, and periodic sequences. Recurrence relations are introduced as formulas that generate the terms of a sequence. The concept of a general term is explained as a way to define the nth term of a sequence. Finally, the document defines a series as the sum of the terms of a sequence, and introduces sigma notation as a way to write series in a condensed form.
1. 30: Sequences and Series30: Sequences and Series
ยฉ Christine Crisp
โโTeach A Level MathsโTeach A Level Mathsโ
Vol. 1: AS Core ModulesVol. 1: AS Core Modules
2. Sequences and Series
Module C1
AQAEdexcel
OCR
MEI/OCR
Module C2
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3. 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 โโ
A sequence is an ordered list of numbers
The 3 dots are used to show that a sequence continues
4. Sequences and Series
Recurrence 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 1+nu
is the 1st
term, so1u 31 =u
5232 =+=โ u
7253 =+=โ u
etc.
โ= 1n 212 += uu
โ= 2n
11
5. Sequences and Series
Recurrence Relations
nn uu 41 โ=+
e.g. 1 Give the 1st
term and write down a
recurrence relation for the sequence
...,64,16,4,1 โโ
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 โ=+
Recurrence relation:
A formula such as is called a
recurrence relation
21 +=+ nn uu
6. Sequences and Series
Recurrence Relations
e.g. 2 Write down the 2nd
, 3rd
and 4th
terms of the
sequence given by 32,5 11 โ== + ii uuu
โ= 1iSolution: 32 12 โ= uu
73)5(22 =โ=uโ
โ= 2i 32 23 โ= uu
113)7(23 =โ=uโ
โ= 3i 32 34 โ= uu
193)11(24 =โ=uโ
The sequence is ...,19,11,7,5
7. Sequences and Series
Properties of sequences
Convergent sequences approach a certain value
e.g. approaches 2...1,1,1,1,1 16
15
8
7
4
3
2
1
n
nu
8. Sequences and Series
Properties of sequences
e.g. approaches 0...,,,,,1 16
1
8
1
4
1
2
1 โโ
This convergent sequence also oscillates
Convergent sequences approach a certain value
n
nu
10. Sequences and Series
Properties of sequences
e.g. ...,16,8,4,2,1 โโโ
This divergent sequence also oscillates
Divergent sequences do not converge
n
nu
11. Sequences and Series
Properties of sequences
...,3,2,1,3,2,1,3,2,1
This divergent sequence is also periodic
Divergent sequences do not converge
n
nu
12. Sequences and Series
Convergent Values
It is not always easy to see what value a sequence
converges to. e.g.
n
n
n
u
u
uu
310
,1 11
โ
== +
...,
11
103
,
7
11
,7,1 โโThe sequence is
To 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
u
u
310 โ
=
01032
=โ+โ uu
0)2)(5( =โ+โ uu 25 โ โ=โ uu since
uu 3102
โ=
Multiply by u :
13. Sequences and SeriesExercises
1. 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
u
uu
1
2 11 =โ= +and
2. What value does the sequence given by ,u 21 =
34 11 +=โ= + nn uuu and(a)
nn uuu 2
1
11 16 โ== +and(c)
Ans: 8,5,2,1,4 โโ Divergent
Ans: 2,,2,,2 2
1
2
1 โโโโโ Divergent Periodic
Ans: 1,2,4,8,16 โโ Convergent Oscillating
uuu nn ==+1Let
370330 =โ โ+โ =โ uuu
7
30
=โ u
to?converge3301 +โ =+ nn uu
14. Sequences and Series
General Term of a Sequence
Some 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 =โโ nu...,64,16,4,1 1
)4( โ
โ n
15. Sequences and Series
Exercises
Write out the first 5 terms of the following
sequences
1.
(b)
n
nu )2(โ=
nun 41โ=(a)
2
2nun =(c)
n
nu )1(โ=(d)
19,15,11,7,3 โโโโโ
32,16,8,4,2 โโโ
50,32,18,8,2
1,1,1,1,1 โโโ
Give the general term of each of the following sequences2.
...,7,5,3,1(a) 12 โ= nun
...,243,81,27,9,3 โโโ(c)
(b) ...,25,16,9,4,1
(d) ...,5,5,5,5,5 โโ 5)1( 1+
โ= n
nu
2
nun =
n
nu )3(โ=
16. Sequences and Series
Series
When the terms of a sequence are added, we get a
series
...,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.
โ
10
1
2
ncan be written
โ is the Greek capital letter S, used for Sum
1st
value of n
last value of n
17. Sequences and Series
16...8642 +++++(a) โ=
8
1
2n
100
3...2793 ++โ+โ=(b)
2. Write the following using sigma notation
Exercises
1. Write out the first 3 terms and the last term of
the series given below in sigma notation
โ โ
20
1
12n(a) += 1
1024...842 ++++(b) ( )โ=
10
1
2
n
+3
n = 1n = 2
39...5 ++
( )โ โ
100
1
3
n
n = 20
