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        3        4
,         ,         ,         , ...
2        3        4        5

We see that as n becomes large the numbers approach 1.  In particular if any small error number e is given, we can find an N such that for n > N, |an -1| < e.  We say that the limit of the sequence approaches 1

In general,

If an is a sequence that converges to a limit L then for any e > 0

we can find an N such that for all n > N

|an - L| < e

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 + 1
an  =
n - 3

by considering the function

2n + 1
f(n)  =
n - 3

We note 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.

The Squeeze Theorem

Suppose that

lim an  =  lim bn  =  L

and that there is an N such that for any n > N,

an   <   cn   <   bn

then

lim cn = L

Example

Show that

Note that

-1             sin n            1
<                <
n                n!              n

both the left hand and right hand sides converge to 0 hence

Monotonic and Bounded Sequences

 Definition of Monotonicity and Boundedness A sequence is monotonically decreasing (increasing) if           an  >  an+1   (an  <  an+1 )  for all n. A sequence is bounded from above (below) if there is a number M such that           an < M  ( an > M) for all n.

Example

Determine the montonicity and boundedness of the following sequences

1. cos(n)

2. 1/n3

Solution

1. cos(n) is not monotonic since, for example

cos 1  >  cos 2

and

cos 3  <  cos 4

However, cos(n) is bounded above by 1 and below by -1 since

-1  <  cos(n)  <  1

2. 1/n3 is monotonic since for n > 0

(1/n3)' = -3/n4  <  0

so 1/n3 is monotonically decreasing.

Exercises:

Classify the monotonicity and boundedness of the following sequences:

1. an = sin(n)

2. an = 1/n

3.            n + 1
an  =
n + 2

4. an = n2 + 1

 Theorem A bounded monotonic sequence converges

Example

Show that that sequence

n
an =
en

converges.

Solution

If

x
f(x)  =             =  xe-x
ex

Then

f '(x) = (1 - x) e-x

Which is always negative for x > 1.  Hence an is monotonic.

for x > 0,

xe-x  <  1

so we can conclude that the sequence converges.