8.1: Sequences

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.

A sequence is defined explicitly if there is a formula that allows you to find individual terms independently.

2

1

1

n

na n

Example:

To find the 100th term, plug 100 in for n:

100

100 2

1

100 1a

1

10001

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

You have to keep going this way until you get the term you need.

An arithmetic sequence has a common difference between terms.

Arithmetic sequences can be defined recursively:

3d Example: 5, 2, 1, 4, 7, ...

ln 6 ln 2d ln 2, ln 6, ln18, ln 54, ...6

ln2

ln 3

or explicitly: 1 1na a d n

1n na a d

st term11 a

An geometric sequence has a common ratio between terms.

Geometric sequences can be defined recursively:

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

2

10

10r

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

or explicitly:

1st term1 a

raa nn 1

11

nn raa

Example: 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

You can determine if a sequence converges by finding the limit as n approaches infinity.

Does converge?2 1

n

na

n

2 1limn

n

n

The sequence converges and its limit is 2.

(L’Hopital)

1

2lim

n

• If the sequence has a limit as n approaches infinity, it converges.

• If the sequence does not have a limit, we say it diverges.

Absolute Value Theorem for Sequences

If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero.

Sandwich Theorem for Sequences

If and there is an integer N for which an ≤ bn ≤ cn for all n > N, then

Lca nn

nn

limlimLbn

n

lim

• The calculator can graph sequences defined recursively or explicitly.

Graph a sequence defined explicitly by an = (1/2)n

1. Select Seq and Dot modes

4. Press Graph

3. Press Window and enter the following values:

nMin = 1 PlotStart = 1 xmin = 0 ymin = 0

nMax = 10 PlotStep = 1 xmax = 10 ymax = 1

xscl = 1 yscl = 1

2. Enter the expression into y =

Graph a sequence defined recursively by

an = (1/2)an–1

a1 = (1/2)

1. Select Seq and Dot modes

4. Press Graph

3. Press Window and enter the following values:

nMin = 1 PlotStart = 1 xmin = 0 ymin = 0

nMax = 10 PlotStep = 1 xmax = 10 ymax = 1

xscl = 1 yscl = 1

2. Enter the expression into y =

(the condition a1 = (1/2) goes into u(nMin)