Sequences Objective: To develop some of the basic ideas concerning sequences of numbers.

Sequences

Objective: To develop some of the basic ideas concerning

sequences of numbers.

Sequence

• Stated informally, an infinite sequence, or more simply a sequence, is an unending succession of numbers, called terms. It is understood that the terms have a definite order; that is, there is a first term a1, a second term a2, a third term a3, and so forth. Such a sequence would typically be written as

,...,, 321 aaa

Sequence

• Some specific examples are:

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

1,2

1,1

,....6,4,2 ,....1,1,1,1

Sequence

• Each of those sequences has a definite pattern that makes it easy to generate additional terms if we assume that those terms follow the same pattern as the displayed terms. However, such patterns can be deceiving, so it is better to have a rule or formula for generating the terms. One way of doing this it to look for a function that relates each term in the sequence to its term number.

Sequence

• For example, in the sequence each term is twice the term number; that is, the nth term in the sequence is given by the formula 2n. We denote this by writing the sequence as

,...2,...,8,6,4,2 n

,....6,4,2

Sequence

• For example, in the sequence each term is twice the term number; that is, the nth term in the sequence is given by the formula 2n. We denote this by writing the sequence as

• We call the function the general term of this sequence. Now, if we want to know a specific term in the sequence, we need only substitute its term number in the formula for the general term.

,...2,...,8,6,4,2 n

,....6,4,2

nnf 2)(

Example 1

• In each part, find the general term of the sequence.

,...5

4,4

3,3

2,2

1)(a ,...

16

1,8

1,4

1,2

1)(b

,...5

4,4

3,3

2,2

1)( c ,...7,5,3,1)(d

Example 1


,...5

4,4

3,3

2,2

1)(a ,...

16

1,8

1,4

1,2

1)(b

,...5

4,4

3,3

2,2

1)( c ,...7,5,3,1)(d

,...1

,...5

4,4

3,3

2,2

1)(

nn

a

Example 1


,...5

4,4

3,3

2,2

1)(a ,...

16

1,8

1,4

1,2

1)(b

,...5

4,4

3,3

2,2

1)( c ,...7,5,3,1)(d

,...2

1,...,

16

1,8

1,4

1,2

1)(

nb

Example 1


,...5

4,4

3,3

2,2

1)(a ,...

16

1,8

1,4

1,2

1)(b

,...5

4,4

3,3

2,2

1)( c ,...7,5,3,1)(d

,...1

)1(,...,5

4,4

3,3

2,2

1)( 1

n

nc n

Example 1


,...5

4,4

3,3

2,2

1)(a ,...

16

1,8

1,4

1,2

1)(b

,...5

4,4

3,3

2,2

1)( c ,...7,5,3,1)(d

,...12,...,7,5,3,1)( nd

Notation

• When using a general term, there is no need to write the initial terms. It is common only to write the general term enclosed in braces. For the past example, we would write:

11)(

nn

na

12

1)(

nn

b

1

1

1)1()(

n

n

n

nc

112)( nnd

Definition

• 10.1.1 A sequence is a function whose domain is a set of integers. Specifically, we will regard the expression to be an alternative notation for the function

,...3,2,1,)( nanf n

1nna

Graphs of Sequences

• Since sequences are functions, it makes sense to talk about the graph of a sequence. For example, the graph of the sequence is the graph of the equation .,...3,2,1,/1 nny

1/1 nn

Graphs of Sequences

• Since sequences are functions, it makes sense to talk about the graph of a sequence. For example, the graph of the sequence is the graph of the equation .

• Because the right side of this equation is defined only for positive integer values of n, the graph consists of a succession of isolated points.

,...3,2,1,/1 nny

1/1 nn

Graphs of Sequences

• Since sequences are functions, it makes sense to talk about the graph of a sequence. For example, the graph of the sequence is the graph of the equation .

• Because the right side of this equation is defined only for positive integer values of n, the graph consists of a succession of isolated points. Compare to the function

,...3,2,1,/1 nny

1/1 nn

Limit of a Sequence

• Since sequences are functions, we can inquire about their limits. However, because a sequence is only defined for integer values of n, the only limit that makes sense is the limit of as . na

1nna

n

Limit of a Sequence

• Since sequences are functions, we can inquire about their limits. However, because a sequence is only defined for integer values of n, the only limit that makes sense is the limit of as .

• The terms in the sequence increase without bound.

na

1nna

n

1n

Limit of a Sequence


• The terms in the sequence oscillate between -1 and 1.

na

1nna

n

1)1( n

Limit of a Sequence


• The terms in the sequence increase toward a “limiting value” of 1.

na

1nna

n

1/ nn

Limit of a Sequence


• The terms in the sequence also tend toward a “limiting value” of 1, but do so in an oscillatory fashion.

na

1nna

n

n)2/1(1

Limit of a Sequence

• Informally speaking, the limit of a sequence is intended to describe how behaves as . To be more specific, we will say that a sequence approaches a limit L if the terms in the sequence eventually become arbitrarily close to L. Geometrically, this means that for any positive number there is a point

in the sequence after which all terms lie between the lines y = L + and y = L - .

nana n

na

Definition

• 10.1.2 A sequence is said to converge to the limit L if given any , there is a positive integer N such that for n > N. In this case we write

• A sequence that does not converge to some finite limit is said to diverge.

na

Lann

lim

|| Lan

0

Example 2

• The first two sequences below diverge, the second two converge to 1.

Theorem

• 10.1.3 Suppose that the sequences and converge to a limit L1 and L2 respectively, and c is a constant. Then:

na nb

Example 3

• In each case, determine whether the sequence converges or diverges. If it converges, find the limit.

1

1

12)1()(

n

n

n

nb

112)(

nn

na

1

1 1)1()(n

n

nc

128)( nnd

Example 3

• In each case, determine whether the sequence converges of diverges. If it converges, find the limit.

1

1

12)1()(

n

n

n

nb

2

1

12)(

1

nn

na

1

1 1)1()(n

n

nc

128)( nnd

Example 3


divergesn

nb

n

n

1

1

12)1()(

2

1

12)(

1

nn

na

1

1 1)1()(n

n

nc

128)( nnd

Example 3


divergesn

nb

n

n

1

1

12)1()(

2

1

12)(

1

nn

na

01

)1()(1

1

n

n

nc

128)( nnd

Example 3


divergesn

nb

n

n

1

1

12)1()(

2

1

12)(

1

nn

na

01

)1()(1

1

n

n

nc divergesnd n ;28)( 1

Example 4


,...2,...,2,2,2,1)( 32 nb

,...2

1,...,

2

1,

2

1,2

1,1)(

32 na

Example 4


,...2,...,2,2,2,1)( 32 nb

,...2

1,...,

2

1,

2

1,2

1,1)(

32 na 0

2

1lim

nn

n

n2lim

Example 5

• Find the limit of the sequence

1nne

n

Example 5


• This is of the indeterminate form of type We will look at this the limit of the quotient When we apply L’Hopital’s Rule, we have

1nne

n

/xex /

01

limlim xnxn ee

x

Example 5


• This is of the indeterminate form of type We will look at this the limit of the quotient When we apply L’Hopital’s Rule, we have

• We also know that accelerates faster than x.

1nne

n

/xex /

01

limlim xnxn ee

x

xe

Example 7

• The sequence converges to 0 since the even numbered terms and the odd numbered terms both converge to 0.

,...3

1,

2

1,

3

1,

2

1,3

1,2

13322

Example 7

• The sequence converges to 0 since the even numbered terms and the odd numbered terms both converge to 0.

• The sequence diverges since the odd-numbered terms converge to 1 and the even-numbered terms converge to 0.

,...3

1,

2

1,

3

1,

2

1,3

1,2

13322

,....1,4

1,1,

3

1,1,

2

1,1

Theorem

• 10.1.6 If then ,0||lim n

na 0lim

nn

a

Example 9

• Consider the sequence

• If we take the absolute value of each term, we obtain the sequence converges to 0, so

nn

2

1)1,...(

2

1,

2

1,2

1,1

32

,...2

1,...

2

1,

2

1,2

1,1

32 n

02

1)1(lim

nn

n

Homework

• Pages 633-634

• 1-29 odd

Sequences Objective: To develop some of the basic ideas concerning sequences of numbers.

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