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Sequences and Series

Sequences and Series Examples of Sequences e.g. 1 e.g. 2

2, 4, 6, 8, . . .
1 1 1 1, , , , . . . 2 3 4

e.g. 3

1, ? 4, 16, ? 64, . . .
A sequence is an ordered list of numbers

The 3 dots are used to show that a sequence continues

Sequences and Series Recurrence Relations Can you predict the next term of the sequence 3, 5, 7, 9, . . . ? 11 Suppose the formula continues by adding 2 to each term.

The formula that generates the sequence is then

u n ?1 ? u n ? 2
where

u n and u n ?1 are terms of the sequence u1 is the 1st term, so u1 ? 3 n ? 1 ? u2 ? u1 ? 2 ? u2 ? 3 ? 2 ? 5 n ? 2 ? u3 ? u2 ? 2 ? u3 ? 5 ? 2 ? 7
etc.

Sequences and Series Recurrence Relations

A formula such as
recurrence relation

un?1 ? un ? 2 is called a

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

recurrence relation for the sequence

1, ? 4, 16, ? 64, . . .
Solution: 1st term: Recurremce relation:

u1 ? 1 un?1 ? ?4 un

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

a k ?1 ? ?4 a k

Sequences and Series Recurrence Relations e.g. 2 Write down the 2nd, 3rd and 4th terms of the sequence given by Solution:

u1 ? 5,

u i ? 1 ? 2u i ? 3

i ?1 i?2 i?3

? ? ?

u 2 ? 2u 1 ? 3 u 2 ? 2(5) ? 3 ? 7 u 3 ? 2u 2 ? 3 u 3 ? 2(7) ? 3 ? 11
u 4 ? 2u 3 ? 3

?
? ?

The sequence is

u 4 ? 2(11) ? 3 ? 19 5, 7, 11, 19, . . .

Sequences and Series Properties of sequences Convergent sequences approach a certain value e.g. 1, 1 1 , 1 3 , 1 7 , 1 15 . . .
2 4 8 16

approaches 2

un

n

Sequences and Series Properties of sequences Convergent sequences approach a certain value e.g. 1, ? 1 , 1 , ? 1 , 1 , . . .
2 4 8 16

approaches 0

un

n

This convergent sequence also oscillates

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.
un

2, 4, 6, 8, 10, . . .

n

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.

?1, 2, ? 4, 8, ? 16, . . .
un

n

This divergent sequence also oscillates

Sequences and Series Properties of sequences Divergent sequences do not converge e.g.

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

n

This divergent sequence is also periodic

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

2, 4, 6, 8, . . .

u n ? 2n

1 1 1 1 1, , , , . . . un ? 2 3 4 n

1, ? 4, 16, ? 64, . . . un ? ( ?4) n?1

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

Sequences and Series Exercises 1. Write out the first 5 terms of the following sequences (a) (b) (c) (d)

un ? 1 ? 4n
un ? ( ?2) n u n ? 2n 2 un ? (?1) n

?3, ?2, 2, ?1,

? 7, ? 11, ? 15, ? 19 4, ? 8, 16, ? 32 8, 18, 32, 50 1, ? 1, 1, ? 1

2. Give the general term of each of the following sequences (a) 1, 3, 5, 7, . . . un ? 2n ? 1 (b) 1, 4, 9, 16, 25, . . . u ? n2 (c) ?3, 9, ? 27, 81, ? 243, . . .
n

un ? (?3) n

Series When the terms of a sequence are added, we get a series The sequence 1, 4, 9, 16, 25, . . .
gives the series 1 ? 4 ? 9 ? 16 ? 25 ? . . . Sigma Notation for a Series

Sequences and Series

A series can be described using the general term e.g. 1 ? 4 ? 9 ? 16 ? 25 ? . . . ? 100 can be written

?
1

10

n2

last value of n 1st value of n

? is the Greek capital letter S, used for Sum

Sequences and Series Exercises 1. Write out the first 3 terms and the last term of the series given below in sigma notation (a)

? 2n ? 1
1
100 1

20

? 1 ? 3 ? 5 ? . . . ? 39 n =n 1= 2 n = 20
? ? 3 ? 9 ? 27 ? . . . ? 3100

(b)

n ? ? ? ?3

2. Write the following using sigma notation (a) 2 ? 4 ? 6 ? 8 ? . . . (b) 2 ? 4 ? 8 ? . .

? 2n . ? 1024 ? ? ?2?
? 16 ?
10 1 n 1

8

Sequences and Series


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