Sequences, Series, and the Binomial Formula

Lecture 6

2

Section 6.1Sequences

3

SequenceA sequence is a set of numbers arranged

in some order.Each number is labeled with a variable, such as

a.The variable is indexed with a natural number that

tells its position in the sequence.

The numbers a1, a2, a3,. . . are the terms of the sequence.The first term in the sequence is a1, the second term

a2, the third term a3, and so on.

4

General Term of a Sequence

Many sequences follow some sort of pattern.The pattern is usually described by the nth term of

the sequence.This term, an, is called the general term of the

sequence.

5

Finite Sequence

A finite sequence has a specific number of terms and so it has a last term.

An infinite sequence does not have a last term.

6

Sequence Notation

The notation {an} is often used to represent a

sequence whose nth term is an.

The { } indicate that it is a sequence.

Example – sequence notationFind the first five terms of the sequence

7

2 3n

21 1 3 1 3 2a

22 2 3 4 3 1a

23 3 3 9 3 6a

24 4 3 16 3 13a

25 5 3 25 3 22a

8

Recursion Formula

A recursion formula defines a sequence in terms of one or more previous terms.A sequence that is specified by giving the first

term, or the first few terms, and a recursion formula is said to be defined recursively.

9

Section 6.2 Arithmetic and Geometric Sequences

10

Arithmetic SequenceAn arithmetic sequence, or arithmetic

progression, is a sequence where each term is obtained from the preceding term by adding a fixed number called the common difference.If the common difference is d, then an

arithmetic sequence follows the recursion formula an = an-1 + d.

11

Terms of an Arithmetic Sequence

If a1 is the first term of an arithmetic sequence, an the nth term and d the common difference,

then an = a1 + (n - 1)d.

If an-1 and an are consecutive terms of an arithmetic sequence, then d = an - an-1.

12

Geometric SequenceA geometric sequence, or geometric

progression, is a sequence where each term is obtained by multiplying the preceding term by a fixed number called the common ratio.If the common ratio is r, then a geometric

sequence follows the recursion formula: an = ran-1.

13

Terms of a Geometric SequenceIf a1 is the first term of a geometric

sequence, an the nth term, and r the common ratio, then an = rn-1a1

If an-1 and an are consecutive terms of a geometric sequence, then

1

n

n

a

ar

Example – Terms of a SequenceIf the first term is 5, the last (nth) term is

-139, and d = -6, how many terms are there?

14

1 ( 1)na a n d

There are 25 terms.

139 5 ( 1)( 6)n 144 6 6n

150 6 ; 25n n

15

Section 6.3 Series

16

Series

The sum of the terms of a sequence is called a series.The series a1 + a2 + a3 + a4 + a5 is a finite

series with five terms.

17

Infinite Series

A series of the form a1 + a2 + a3 + a4 + · · ·is an infinite series.An infinite series has an infinite or endless

number of terms.

18

Summation NotationSummation or sigma notation means

sum. a indicates Here -

n

n

kk aaaaa 321

1

19

Limits of Summation

The letter k in is called an index of summation.

The summation begins with k = 1 as is indicated below the and ends with k = n as indicated above the .

The numbers below and above theare the limits of summation.

n

kka

1

20

Property 1 of Summation Notation

If a is a constant then

n

k

n

k

kaak11

21

Property 2 of Summation Notation

If a is a constant then

naan

k

1

22

Property 3 of Summation NotationIf x and y are constants then

n

k

n

k

n

k

yxyx1 11

)(

23

Partial Sum

If Sn = a1 + a2 + a3 + · · · + an =then the number Sn is called the nth partial sum of the series.

The sequence S1, S2, S3, . . . , Sn is called the sequence of partial sums.

n

kka

1

24

Arithmetic Series

An arithmetic series is a series formed by the partial sums of an arithmetic sequence.

25

Sum of First n Terms of an Arithmetic SequenceThe sum Sn of the first n terms of an

arithmetic sequence is

where a1 is the first term and an is the nth term.

2

)( 1 nn

aanS

26

Geometric Series

A geometric series is a series formed by the partial sums of a geometric sequence.

27

Sum of First n Terms of a Geometric Sequence

The sum Sn of the first n terms of a geometric sequence is

where a1 is the first term and r is the common ratio.

r

raS

n

n

1

11

28

Compound Interest

A = P(1 + i)n

where A is the amount after n interest periods, P is the principal or initial amount invested, and i is the interest rate per interest period expressed as a decimal.

Example-Summation Notation

Evaluate

29

3

0

2

3 5

k

k k

3

0

2

3 5

k

k k

0 1 2 32 2 2 2

3 0 5 3 1 5 3 2 5 3 3 5

1 2 4 8

0 5 3 5 6 5 9 5

11 4 2

5

445

30

Section 6.4 Infinite Geometric Series

31

Infinite Series, 1

A series that does not have a last term is called an infinite series.

32

Sequence of Partial Sums

series. infinite the of sum the is that

say then we, call will wethat number some

gapproachin be to seems larger, and larger

gets as that, happens it If

is The

series the of the called is

S

S

S

nSSSS

a

nS

n

n

k k

n

.,,,,

.

321

1

sums partial of sequence

sum partial th

33

Convergent Series

If the partial sums of an infinite series approach a finite limit, we say that the series converges or is a convergent series.

34

Divergent Series

A series that does not converge is said to diverge or to be a divergent series.All arithmetic series diverge.

Example – Infinite SeriesFind the fraction that has the repeating decimal form 0.232323 . . . .:

35

0

0.23(0.01)n

n

This decimal can be thought of as the series 0.23 + 0.0023 + 0.000023 +…, which is the geometric series

In this series, so 1 0.23, 0.01,a r 1

1

as

r

0.23 0.23 23

1 0.01 0.99 99

Thus, decimal 0.232323 . . is equivalent to the fraction23

99

36

Sum of an Infinite Geometric Series

diverges. series the thenIf

sum thehas

seriesgeometric infinite the thenIf

,1

.1

,1

1

13

12

111

r

r

aS

rarararaa

r

n

37

Section 6.5 The Binomial Theorem

38

Expansions of x + ySome expansions of x + y are

(x + y)0 = 1(x + y)1 = x + y(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

(x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

39

Pascal’s Triangle

40

Binomial Formula, 1

nn

nnnn

yyxnnn

yxnn

ynxxyx

33

221

32

)2()1(2

)1()(

41

n Factorial

n! = n(n - 1)(n - 2)(n - 3) · · · (3)(2)(1)

42

r

n

)!(!

!

rnr

n

represents

r

n

Example - Factorials

Determine 5! And 8!

43

5! 5 4 3 2 1 120

8! 8 7 6 5 4 3 2 1 40320

44

Binomial Formula, 2 (1 of 2)

rrnn

r

nn

n

nnnn

yxr

n

yxyn

n

yxn

yxn

yxn

xyx

0

1

33

221

1

3

21)(

45

Binomial Formula, 2 (2 of 2)

!

)1()2()1(

!)!(

!

r

rnnnn

rrn

n

r

n

is term general thewhere

46

Binomial Series

32

132

!3

)2()1(

!2

)1(11

.1

1321)1(

xnnn

xnn

nxx)(

nx

n

xxn

nx

nx

nnxx

n

nnn

series. infinite the

get younumber, rational a or negative isIf

if number realany for valid is series binomial The