Sequences and SeriesSequences and Series

PART 1 – Notation +PART 1 – Notation +

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

?The formula continues by adding 2 to each term.

The formula that generates the sequence is then

21 nn uu

223 uu

;31 u

5232 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 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.

uuulet nn 1 u

uu

310

25 uu since

Solve

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

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