Let’s start off this section with a discussion of just what
a sequence is. A sequence is nothing
more than a list of numbers written in a specific order. The list may or may not have an infinite
number of terms in them although we will be dealing exclusively with infinite
sequences in this class. General
sequence terms are denoted as follows,
Because we will be dealing with infinite sequences each term
in the sequence will be followed by another term as noted above. In the notation above we need to be very
careful with the subscripts. The
subscript of 
denotes the next term in the sequence and NOT
one plus the nth
term! In other words,
so be very careful when writing subscripts to make sure that
the “+1” doesn’t migrate out of the subscript!
This is an easy mistake to make when you first start dealing with this
kind of thing.
There is a variety of ways of denoting a sequence. Each of the following are equivalent ways of
denoting a sequence.
In the second and third notations above an is usually given by a formula.
A couple of notes are now in order about these
notations. First, note the difference
between the second and third notations above.
If the starting point is not important or is implied in some way by the
problem it is often not written down as we did in the third notation. Next, we used a starting point of in the third notation only so we could write
one down. There is absolutely no reason
to believe that a sequence will start at . A sequence will start where ever it needs to
start.
Let’s take a look at a couple of sequences.
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Example 1 Write
down the first few terms of each of the following sequences.
(a) [Solution]
(b) [Solution]
(c) ,
where [Solution]
Solution
(a)
To get the first few sequence terms here all we need to do
is plug in values of n into the
formula given and we’ll get the sequence terms.
Note the inclusion of the “…” at the end! This is an important piece of notation as
it is the only thing that tells us that the sequence continues on and doesn’t
terminate at the last term.
[Return to Problems]
(b)
This one is similar to the first one. The main difference is that this sequence
doesn’t start at .
Note that the terms in this sequence alternate in
signs. Sequences of this kind are
sometimes called alternating sequences.
[Return to Problems]
(c) ,
where
This sequence is different from the first two in the sense
that it doesn’t have a specific formula for each term. However, it does tell us what each term
should be. Each term should be the nth digit of π. So we know that
The sequence is then,
[Return to Problems]
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In the first two parts of the previous example note that we
were really treating the formulas as functions that can only have integers
plugged into them. Or,
This is an important idea in the study of sequences (and
series). Treating the sequence terms as
function evaluations will allow us to do many things with sequences that
couldn’t do otherwise. Before delving
further into this idea however we need to get a couple more ideas out of the
way.
First we want to think about “graphing” a sequence. To graph the sequence we plot the points as n
ranges over all possible values on a graph.
For instance, let’s graph the sequence . The first few points on the graph are,
The graph, for the first 30 terms of the sequence, is then,
This graph leads us to an important idea about
sequences. Notice that as n increases the sequence terms in our sequence, in this case, get closer and closer to zero. We then say that zero is the limit (or sometimes the limiting value) of the sequence and
write,
This notation should look familiar to you. It is the same notation we used when we
talked about the limit of a function. In
fact, if you recall, we said earlier that we could think of sequences as
functions in some way and so this notation shouldn’t be too surprising.
Using the ideas that we developed for limits of functions we
can write down the following working
definition for limits of sequences.
Working Definition of Limit
- We
say that
if we can make an as close to L as we want for all sufficiently
large n. In other words, the value of the an’s approach L as n approaches infinity.
- We
say that
if we can make an as large as we want for
all sufficiently large n. Again, in other words, the value of the an’s get larger and larger
without bound as n approaches
infinity.
- We
say that
if we can make an as large and negative as
we want for all sufficiently large n. Again, in other words, the value of the an’s are negative and get
larger and larger without bound as n
approaches infinity.
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The working definitions of the various sequence limits are
nice in that they help us to visualize what the limit actually is. Just like with limits of functions however, there
is also a precise definition for each of these limits. Let’s give those before proceeding
Precise Definition of Limit
We won’t be using the precise definition often, but it will
show up occasionally.
Note that both definitions tell us that in order for a limit
to exist and have a finite value all the sequence terms must be getting closer
and closer to that finite value as n
increases.
Now that we have the definitions of the limit of sequences
out of the way we have a bit of terminology that we need to look at. If exists and is finite we say that the sequence
is convergent. If doesn’t exist or is infinite we say the
sequence diverges. Note that sometimes we will say the sequence diverges to if and if we will sometimes say that the sequence diverges to .
Get used to the terms “convergent” and “divergent” as we’ll
be seeing them quite a bit throughout this chapter.
So just how do we find the limits of sequences? Most limits of most sequences can be found
using one of the following theorems.
Theorem 1
This theorem is basically telling us that we take the limits
of sequences much like we take the limit of functions. In fact, in most cases we’ll not even really
use this theorem by explicitly writing down a function. We will more often just treat the limit as if
it were a limit of a function and take the limit as we always did back in
Calculus I when we were taking the limits of functions.
So, now that we know that taking the limit of a sequence is
nearly identical to taking the limit of a function we also know that all the
properties from the limits of functions will also hold.
Properties
These properties can be proved using Theorem 1 above and the
function limit properties
we saw in Calculus I or we can prove them directly using the precise definition
of a limit using nearly identical proofs
of the function limit properties.
Next, just as we had a Squeeze Theorem for function limits
we also have one for sequences and it is pretty much identical to the function
limit version.
Squeeze Theorem for
Sequences
Note that in this theorem the “for all for some N”
is really just telling us that we need
to have for all sufficiently large n, but if it isn’t true for the first
few n that won’t invalidate the
theorem.
As we’ll see not all sequences can be written as functions
that we can actually take the limit of.
This will be especially true for sequences that alternate in signs. While we can always write these sequence
terms as a function we simply don’t know how to take the limit of a function
like that. The following theorem will help
with some of these sequences.
Theorem 2
Note that in order for this theorem to hold the limit MUST
be zero and it won’t work for a sequence whose limit is not zero. This theorem is easy enough to prove so let’s
do that.
Proof of Theorem 2
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The main thing to this proof is to note that,
Then note that,
We then have and so by the Squeeze Theorem we must also
have,
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The next theorem is a useful theorem giving the
convergence/divergence and value (for when it’s convergent) of a sequence that
arises on occasion.
Theorem 3
Here is a quick (well not so quick, but definitely simple)
partial proof of this theorem.
Partial Proof of
Theorem 3
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We’ll do this by a series of cases although the last case
will not be completely proven.
Case 1 :
We know from Calculus I that if and so by Theorem 1 above we also know that and so the sequence diverges if .
Case 2 :
In this case we have,
So, the
sequence converges for and in this case its limit is 1.
Case 3 :
We know from Calculus I that if and so by Theorem 1 above we also know that and so the sequence converges if and in this case its limit is zero.
Case 4 :
In this case we have,
So, the
sequence converges for and in this case its limit is zero.
Case 5 :
First let’s note that if then then by Case 3 above we have,
Theorem 2 above
now tells us that we must also have, and so if the sequence converges and has a limit of 0.
Case 6 :
In this case the sequence is,
and hopefully
it is clear that doesn’t exist. Recall that in order of this limit to exist
the terms must be approaching a single value as n increases. In this case
however the terms just alternate between 1 and -1 and so the limit does not
exist.
So, the
sequence diverges for .
Case 7 :
In this case we’re not going to go through a complete
proof. Let’s just see what happens if
we let for instance. If we do that the sequence becomes,
So, if we get a sequence of terms whose values
alternate in sign and get larger and larger and so doesn’t exist. It does not settle down to a single value
as n increases nor do the terms ALL
approach infinity. So, the sequence
diverges for .
We could do
something similar for any value of r
such that and so the sequence diverges for .
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Let’s take a look at a couple of examples of limits of
sequences.
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Example 2 Determine
if the following sequences converge or diverge. If the sequence converges determine its
limit.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
(a)
In this case all we need to do is recall the method that
was developed in Calculus I to deal with the limits of rational
functions. See the Limits At Infinity, Part I section
of my Calculus I notes for a review of this if you need to.
To do a limit in this form all we need to do is factor
from the numerator and denominator the largest power of n, cancel and then take the limit.
So the sequence converges and its limit is .
[Return to Problems]
(b)
We will need to be careful with this one. We will need to use L’Hospital’s Rule on
this sequence. The problem is that
L’Hospital’s Rule only works on functions and not on sequences. Normally this would be a problem, but we’ve
got Theorem 1 from above to help us out.
Let’s define
and note that,
Theorem 1 says that all we need to do is take the limit of
the function.
So, the sequence in this part diverges (to ).
More often than not we just do L’Hospital’s Rule on the
sequence terms without first converting to x’s since the work will be identical regardless of whether we use
x or n. However, we really
should remember that technically we can’t do the derivatives while dealing
with sequence terms.
[Return to Problems]
(c)
We will also need to be careful with this sequence. We might be tempted to just say that the
limit of the sequence terms is zero (and we’d be correct). However, technically we can’t take the
limit of sequences whose terms alternate in sign, because we don’t know how
to do limits of functions that exhibit that same behavior. Also, we want to be very careful to not
rely too much on intuition with these problems. As we will see in the next section, and in
later sections, our intuition can lead us astray in these problem if we
aren’t careful.
So, let’s work this one by the book. We will need to use Theorem 2 on this
problem. To this we’ll first need to
compute,
Therefore, since the limit of the sequence terms with
absolute value bars on them goes to zero we know by Theorem 2 that,
which also means that the sequence converges to a value of
zero.
[Return to Problems]
(d)
For this theorem note that all we need to do is realize
that this is the sequence in Theorem 3 above using . So, by Theorem 3 this sequence diverges.
[Return to Problems]
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We now need to give a warning about misusing Theorem 2. Theorem 2 only works if the limit is
zero. If the limit of the absolute value
of the sequence terms is not zero then the theorem will not hold. The last part of the previous example is a
good example of this (and in fact this warning is the whole reason that part is
there). Notice that
and yet, doesn’t even exist let alone equal 1. So, be careful using this Theorem 2. You must always remember that it only works
if the limit is zero.
Before moving onto the next section we need to give one more
theorem that we’ll need for a proof down the road.
Theorem 4
Proof of Theorem 4