Sequences

Definition of a Sequence

A sequence is a list of numbers, or more formally, a function f(n) from the natural numbers to the real numbers.  We write

an

to mean the nth term of the sequence.

Example:

If

1
an  =
n + 2

then we have

a1 = 1/3, a2 = 1/4, etc.

Exercise:

Write the general term an for the following sequences:

1.  -1,1,-1,1,-1,1,...

2. 1,4,9,16,...

3. 1/2, -1/6, 1/24,-1/120,...

4. 1, 1/2, -1/4, -1/8, 1/16, 1/32, -1/64, -1/128, ...

The Limit of a Sequence

Consider the sequence

1/2, 2/3, 3/4, 4/5, ...

We see that as n becomes large the numbers approach 1.

If there is no such L then we say that the sequence diverges.

 Theorem              Let f(n) = an  be a sequence, then an -> L if and only if

Example:
We find the limit of the sequence

2n + 3
an =
n - 3

by considering the function

2n + 3
f(n)  =
n - 3

We not that as

n ->

we get

/

hence we can use L'Hopital's Rule:  Taking derivatives of the top and bottom, we have 2/1 hence the limit is 2.

Exercises

Find the limit of the following sequences or state that the sequences diverge.

1. 1/2, 1/4, 1/8, 1/16, ...

2. 5/6, 6/8, 7/10, 8/12, .

3. 1/2, 2/5, 4/8, 8/16, ...

4. 1/cos(1), 2/cos(2), 3/cos(3), ...

Application

Currently there are 5,000 of a certain species of monkeys left in a jungle.  Each year, because of poachers and loss of habitat, the population decreases by 5%.  Write a sequence that models this and find the limit of the sequence.

Solution

We have

a0  =  5000        a2  =  5000 - 5000(.05)  =  5000(.95)

a2  =  5000(.95) - 5000(.95)(.05)  =  5000(.95)(.95)  =  5000(.95)2

Following this pattern, we get

an  =  5000(.95)n

We find the limit of this sequence to be 0, that is the population will eventually become extinct.

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