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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 from our
sequence terms, 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