Math 1210-007
Limits of Sequences
Fall 2010
Basic Sequences
Definition: A sequence is an infinitely long, ordered set of numbers. For example, the set of
natural numbers, the set of all odd numbers, the set of all prime numbers, and the set of all
perfect squares are sequences. The notation
is used to describe an infinitely long sequence
where
is the nth number in the sequence. That is
describes the ordered set of numbers
which continues indefinitely.
Examples of Sequences:
Given that
examples:
is a sequence we have a choice of how we can define each
. Here are some
describes the sequence
describes the sequence
describes the sequence
describes the sequence
When analyzing Zeno's paradox we found that the total distance traveled after n steps is
which also describes a sequence and can be denoted
.
The Limit of a Sequence
Many sequences, though not all, appear to get closer and closer to a particular number as one
progresses further and further into the sequence. The number being approached is referred to as
the limit of the sequence and is usually denoted as L. The limit of a sequence is defined more
precisely as follows:
Definition: L is the limit of a sequence
a sufficiently large number N such that
if for any
, no matter how small, there exists
whenever
.
In other words, if I give you a really small number of my choice, called , and you can always
pick a number N large enough so that after the Nth step in the sequence all of the numbers are
less than units away from the limit L, then L is the limit of the sequence.
Examples:
1.
We can show that the limit of the sequence
, where
, is L = 0. To do this we
need to show that for any number
we can find a number N sufficiently large so that
whenever
. We can simplify the expression
as follows:
Therefore we need to satisfy the inequality
we find that
. Therefore any number
. Notice that if we solve this inequality for n
in this sequence will be within
units of the
limit, L, as long as n is larger than . Therefore set
. Since we have a logical way of
determining N, regardless of the choice of , then L = 0 is in fact the limit of this sequence.
2.
It is typically easy to prove the limit of a sequence defined by a first order rational
function (one linear function divided by a second linear function). Let
be the sequence
defined by
. For this sequence it turns out that the limit is L = 5, but let's prove that
this the case. To do this we need to show that for any number
we can find a number N
sufficiently large so that
whenever
. We can simplify the expression
as follows:
Therefore we need to satisfy the inequality
. Notice that if we solve this inequality for n
we find that
in this sequence will be within
. Therefore any number
units of the
limit, L = 5, as long as n is larger than
. Therefore set
. Since we have a
logical way of determining N, regardless of the choice of , then L = 5 is in fact the limit of this
sequence.
3.
We can show that the limit of the sequence
, where
, is
whenever
. To do this we need to show that for any number
we can find a
number N sufficiently large so that
whenever
. We can simplify the
expression
as follows:
Therefore we need to satisfy the inequality
. We can solve this inequality for n in a
series of algebraic steps (follow the steps from left to right).
Note that in the last step we flipped the direction of the inequality because we divided both sides
by
which is a negative number (Recall that the log of a number smaller than 1 is
negative). Therefore any number
in this sequence will be within units of the limit, L, as
long as n is larger than
. Therefore set
logical way of determining N, regardless of the choice of , then
this sequence.
. Since we have a
is in fact the limit of
For some sequences the limiting value is not so obvious. Consider the following two examples:
4.
Let
be the sequence where
. This sequence has two properties about
it that seem to oppose each other. First, notice that the exponent is getting larger and large, and
raising a number greater than 1 to a large power gives a large number. However, keep in mind
that raising 1 to a large power still gives you one. Observe that the quantity inside the
parenthesis gets closer and closer to 1 as n gets larger. So the part inside the parenthesis is
forcing the sequence to converge to 1 while the exponent outside of the parenthesis is forcing the
sequence to be very large. Which of the two opposing forces wins? Or do they meet in the
middle somewhere? It turns out that the limit of this sequence is the number
.
5.
Consider the sequence of regular polygons shown to the
right. Let's suppose that each polygon has a radius of 1 (the radius
of a regular polygon is the distance from its center to one of its
vertices). Let's construct the sequence
so that each is
defined to be the area of the regular n-gon in this sequence of
regular polygons. On can use trigonometry to calculate the area of
each n-gon to be
assuming we are working in terms of radians rather than degrees.
Geometrically we can see that as we go further and further into the sequence of polygons, each
shape appears to be approaching the shape of a circle of radius 1. Thus the area of each polygon,
, must be approaching the area of the circle of radius 1. The area of the circle of radius 1 is .
So it turns out that the sequence
has the limit
.
Methods for Finding the Limit of a Sequence
Finding the limiting value of a sequence is often easier than proving a given value is in fact the
limit of a sequence. Let us first define some notation that is commonly used with limits. Let
be some sequence with limit L. A more concise way of expressing this is as follows:
Common ways to read this expression are: "The value of
approaches L as n becomes large" or
"The limit of
as n becomes large is L". Here are some basic properties of limits that will help
in determining the limit of a sequence.
1.
2.
3.
4.
5.
6.
7.
8.
If
then
In properties 5, 6, and 7 it is assumed that the value of k is a constant that does not depend on n.
In property 8, when we say the limit is positive or negative infinity, what we mean is that the
sequence
is unbounded. More precisely this means that if you give me any number
,
no matter how large, I can find a value N large enough so that
whenever
. So
property 8 essentially says that if a sequence blows up to infinity or negative infinity, then its
reciprocal must vanish to zero. Now let's use these properties to find the limits of some actual
sequences!
Examples:
1.
Notice that we can write this expression equivalently as
. We can consider this as
two different sequences summed together. That is, let
be the sequence defined by
and let
be the sequence defined by
. It is easy to see that
, so if we
can find the limits of the sequences
and
, then we can apply rule 1 to determine the
limit of
. Clearly, by property 6 we know that
. Then we can apply property
8 to
since its denominator goes to infinity as n gets large. Therefore
.
Finally applying rule 1 we have that
2.
We see that as n becomes very large both the numerator and denominator of this fraction become
very large as well. So it is unclear what the limit should be. However, if we multiply the top and
bottom by it makes the limit easier to see. Multiplying the numerator and denominator by
gives
Let's define the sequence
and
to be the numerator and denominator, respectively.
Then we can find the limits for these individual sequences and apply property 4. Let's start with
.
Notice that
is the sum of three different sequences, the first one whose limit is 4 (from
property 6), and the next two whose limits are 0 (from property 8). Thus we see that
Next we'll study
Notice that
is the sum of three different sequences, the first one whose limit is 2 (from
property 6), and the next two whose limits are 0 (from property 8). Thus we see that
So now finally, by applying property 4, we see that
3.
where
.
Given the inequality constraint on r we can divide all sides by r to see that
simply that
or more
. Notice further that
Let us focus on the denominator of this last expression and call this sequence
which is
defined as
. We can see that this sequence is unbounded because is greater than 1
and raising a number that is greater than 1 to a larger and larger power will cause it to increase
without bound. Therefore we can apply property 8 to the sequence
, whose denominator is
unbounded, and determine that its limit must go to zero. That is