Math 1210-003
Computing Limits of Sequences
Spring 2012
1/18/12
Notation
Let us begin by defining the notation that is commonly used with limits. Let
with limit L. A more concise way of expressing this is as follows:
be sequence
The common way to read this expression is "The limit of
as n goes to infinity is L". It is
usually convenient to use the same letter to represent the limit of a sequence. For example, we
can denote the limit of sequence
to be and the limit of sequence
to be etc.
Unbounded Sequences
There is one brief topic we will need to discuss before diving into techniques for
computing the limits of sequences. It is important to know when a sequence is unbounded,
which essentially means that it diverges to or
. In other words, it continues to grow,
without bound. For example, the sequence of positive integers is unbounded because the
numbers continue to grow indefinitely and there is no number in which you could claim is
larger than all numbers in the sequence.
DEFINITION: A sequence,
, is said to be unbounded if for any number
how large, there exists a value N sufficiently large so that
whenever
, no matter
.
Whereas with limits the idea was to find a number which depended on , it is now necessary
to find a number that depends on . No matter which number is chosen for , we need a
systematic way of calculating . In other words, we need to be able to write as a function of
. This is done using the same three step process as was done with verifying limits. To help
illustrate how this words, let's work through one example.
EXAMPLE: Let the sequence
unbounded.
STEP 1 (substitute):
STEP 2 (simplify & solve):
Solving this inequality for
STEP 3 (define N): Let
be defined by
. Show that this sequence is
becomes
simplifies to
gives that
.
, since
is always positive.
Math 1210-003
Computing Limits of Sequences
Spring 2012
1/18/12
Therefore, we see that the sequence
is unbounded because we have a systematic way of
assigning N given any value of . That is, all numbers in the sequence beyond the Nth number
are larger in magnitude than , when
.
Methods for Finding the Limit of a Sequence
There are some sequences for which the limit is obvious. For example, if c is a number,
and the sequence
is defined as
, then clearly
. Since every number
in the sequence is the same, then the limit of that sequence must be equal to all numbers in the
sequence.
Here are some basic properties of limits that are helpful in calculating the limits of more
complicated sequences. For each of these properties, assume that
and
are arbitrary
sequences, and that is an arbitrary constant.
1.
2.
3.
4.
5.
6.
7.
8.
If
is an unbounded sequence and
, then
Each of the above properties provides a means of evaluating the limit of a more complicated
sequence using the limits of more simply defined sequences. For example, property 1 says that
if a sequence can be represented as the sum of two sequences, then the limit of the sequence
original sequence can be calculated as the sum of the two simpler sequences. A few concrete
examples will help clear things up.
Math 1210-003
Computing Limits of Sequences
Spring 2012
1/18/12
EXAMPLES:
1.
Use the properties of limits to calculate the limit of
Observe that
defined be
.
. This could be represented as the sum of two sequences.
Specifically, if
and
then
and
, then rule 1 tells us that the limit of
. Thus, if we can find the limits of
is the sum of those two limits.
From property 6 it is clear that
From property 8 we can see that
since the denominator represents an unbounded sequence. Now
applying rule 1 we see that
2.
Use the properties of limits to calculate the limit of
defined be
.
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 the limit is easier to compute. Multiplying the numerator and
denominator by
gives
Let's define the sequence
and
to be the numerator and denominator of this
expression, 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 a combination of properties 7 and 8).
Thus we see that
Math 1210-003
Computing Limits of Sequences
Spring 2012
1/18/12
Next we'll analyze
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 a combination of properties 7 and 8).
Thus we see that
Finally, applying property 4 yields
3.
Use the properties of limits to calculate the limit of
defined be
.
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 the limit is easier to compute. This yields
Let's define the sequence
and
to be the numerator and denominator of this
expression, respectively. Then we can find the limits for these individual sequences and apply
property 4. Let's start with
.
Notice that
is the sum two different sequences, the first of which is unbounded, and the
next whose limits is 0 (from property 8). Thus we see that
Next we'll analyze
Math 1210-003
Computing Limits of Sequences
Spring 2012
1/18/12
Notice that
is the sum of two different sequences, the first one which 1 (from property 6),
and the next whose limit is 0 (from a combination of properties 7 and 8). Thus we see that
Finally, applying property 4 yields
So we would conclude that this sequence is unbounded, or that it diverges.