9.1Sequences

A sequence is a list of numbers written in an explicit order.

1 2 3,, , ... , , ... n na a a a a

nth term

Any real-valued function with domain a subset of the positive integers is a sequence.

If the domain is finite, then the sequence is a finite sequence.

In calculus, we will mostly be concerned with infinite sequences.

Sequence

The last example is a recursively defined sequence known as the Fibonacci Sequence.

2

2

1

15

{1,4,9,16,25,...}

{1, 1, 2, 3, 5, 8, 13, 21,...}

n

n

n

n

na

n

an

a

Examples

A sequence is defined recursively if there is a formula that relates an to previous terms.

We find each term by looking at the term or terms before it:

1 2 for all 2n nb b n Example: 1 4b

1 4b

2 1 2 6b b

3 2 2 8b b

4 3 2 10b b

A geometric sequence is a sequence in which the ratios between two consecutive terms are the same. That same ratio is called the common ratio.

Geometric sequences can be defined recursively:

2r Example: 1, 2, 4, 8, 16, ...

1n na a r

1

2

10

10r

2 110 , 10 , 1, 10, ... 10

or explicitly: 11

nna a r

If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.

41

1

48

6

a r

a r

3 8r

2r

2 12 1a a r

16 2a

13 a

13 2

n

na

Example

• Let’s take a look at the sequence

• What will happen as n gets large? • If a sequence {an} approach a number L as n

approaches infinity, we will write

and say that the sequence converges to L. • If the limit of a sequence does not exist, then the

sequence diverges.

2n

na

n

lim nn

a L

Limit and Convergence

Does converge?2 1

n

na

n

2 1limn

n

n

2 1limn

n

n n

2 1lim limn n

n

n n

2 0

2

The sequence converges to 2.

Example

Graph the sequence.

• Same as limit laws for functions in chapter 2.• Theorem:

• Squeeze Theorem• Absolute Value Theorem:For the sequence {an},

Properties of Limits

Let f (x) be a function of a real variable such that

If {an} is a sequence such that f (n) = an for every positive integer n, then lim n

na L

lim ( )x

f x L

lim 0, lim 0.n nn nif a then a

23

1

n

nan

nan

3

3

ln

n

nan

n

nc

5

3

!)1( nb nn

n

nbn

1

)2sin(

Examples

Determine the convergence of the following sequences.

• A sequence is called increasing if for all n.

• A sequence is called decreasing if for all n.

• It is called monotonic if it is either increasing or decreasing.

1n na a

Monotonic Sequence

1n na a

• A sequence is bounded above if there is a number M such that an ≤ M for all n.

• A sequence is bounded below if there is a number N such that N ≤ an for all n.

• A sequence is a bounded sequence if it is bounded above and below.

Bounded Sequence

Theorem: Every bounded monotonic sequence is convergent.

2

6

nan

!nan

1

n

nbn

3

4 1000n

nc

n

1)1( nna

Examples

Determine whether the sequence is bounded, monotonic and convergent.

nna )2(