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.
A couple of notes are now in order about these notations.
Example: Write down the first few terms of each of the following sequences.
Solution. 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.
Solution. This one is similar to the first one. The main difference is that this sequence doesn’t start at n=1.
Note that the terms in this sequence alternate in signs. Sequences of this kind are sometimes called alternating sequences.
Solution. 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 p. So we know that π = 3.14159265359…
The sequence is then,
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 we couldn’t do otherwise. Before delving further into this idea however, we need to get a couple more ideas out of the way.
Let's discuss “graphing” a sequence.
To graph the sequence {an}we plot the points (n,an) 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,
Using the ideas that we developed for limits of functions we can write down the following working definition for limits of sequences.
1. 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.
2. 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.
3. 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.
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.
1. We say that = L if for every number ε>0 there is an integer N such that
2. We say that = ∞ if for every number M>0 there is an integer N such that
3. We say that = -∞ if for every number M<0 there is an integer N such that
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.
So just how do we find the limits of sequences? Most limits of most sequences can be found using one of the following theorems.
Given the sequence {an} if we have a function f(x) such that f(n) = an and
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.
If {an} and {bn} are both convergent sequences then,
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.
Note that in this theorem the “for all n>N for some N” is really just telling us that we need to have an≤cn≤bn 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.
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
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,
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.
The sequence converges if −1<r≤1 and diverges for all other values of r. Also,
Here is a quick (well not so quick, but definitely simple) partial proof of this theorem.
Partial Proof of Theorem 3
We’ll do this through a series of cases although the last case will not be completely proven.
Case 1: r>1
We know from Calculus I that and so by Theorem 1 above we also know that and so the sequence diverges if r>1.
Case 2: r=1
In this case, we have,
So, the sequence converges for r=1 and in this case its limit is 1.
Case 3 : 0<r<1
We know from Calculus I that and so by Theorem 1 above we also know that and so the sequence converges if 0<r<1 and in this case its limit is zero.
Case 4: r=0
In this case, we have,
So, the sequence converges for r=0 and in this case its limit is zero.
Case 5: −1<r<0
First, let’s note that if −1<r<0 then 0<|r|<1 then by Case 3 above we have,
Theorem 2 above now tells us that we must also have,and so if −1<r<0 the sequence converges and has a limit of 0.
Case 6: r=−1
In this case, the sequence is,
and hopefully, it is clear that doesn’t exist. Recall that in order for 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 r=−1.
Case 7: r<−1.
In this case, we’re not going to go through complete proof.
Let’s just see what happens if we let r=−2 for instance. If we do that the sequence becomes,
So, if r=−2 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 r=−2.
We could do something similar for any value of r such that r<−1 and so the sequence diverges for r<−1.
Let’s take a look at a couple of examples of limits of sequences.
Example: Determine if the following sequences converge or diverge. If the sequence converges determine its limit.
Solution. 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 3/5.
Solution.
Solution.
Solution. For this theorem note that all we need to do is realize that this is the sequence in Theorem 3 above using r=−1. So, by Theorem 3 this sequence diverges.
Warning - 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 on to the next section we need to give one more theorem that we’ll need for proof down the road.
For the sequence {an} if both and then {an} is convergent and
Proof of Theorem 4
Let ε>0.
Then since there is an N1>0 such that if n>N1 we know that,
Likewise, because there is an N2>0 such that if n>N2 we know that,
Now, let and let n>N. Then either an=a2k for some k>N1 or an=a2k+1 for some k>N2 and so in either case we have that,
Therefore, and so {an} convergent.
In the previous section, we introduced the concept of a sequence and talked about the limits of sequences and the idea of convergence and divergence for a sequence. In this section we want to take a quick look at some ideas involving sequences. Let’s start off with some terminology and definitions.
Given any sequence {an} we have the following.
1. We call the sequence increasing if an<an+1 for every n.
2. We call the sequence decreasing if an>an+1 for every n.
3. If {an} is an increasing sequence or {an} is a decreasing sequence we call it monotonic.
4. If there exist a number m such that m≤an for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.
5. If there exists a number M such that an≤M for every n we say the sequence is bounded above. The number M is sometimes called an upper bound for the sequence.
Note that in order for a sequence to be increasing or decreasing it must be increasing/decreasing for every n. In other words, a sequence that increases for three terms and then decreases for the rest of the terms is NOT a decreasing sequence! Also, note that a monotonic sequence must always increase or it must always decrease.
Before moving on we should make a quick point about the bounds of a sequence that is bounded above and/or below. We’ll make the point about lower bounds, but we could just as easily make it about upper bounds.
A sequence is bounded below if we can find any number m such that m≤an for every n.
Let’s take a look at a couple of examples.
Example 1: Determine if the following sequences are monotonic and/or bounded.
The sequence terms in this sequence alternate between 1 and -1 and so the sequence is neither an increasing sequence nor a decreasing sequence. Since the sequence is neither an increasing nor decreasing sequence it is not a monotonic sequence.
The sequence is bounded, however, since it is bounded above by 1 and bounded below by -1.
Again, we can note that this sequence is also divergent.
This sequence is a decreasing sequence (and hence monotonic) since,
The terms in this sequence are all positive and so it is bounded below by zero. Also, since the sequence is a decreasing sequence the first sequence term will be the largest and so we can see that the sequence will also be bounded above by 2/25. Therefore, this sequence is bounded.
We can also take a quick limit and note that this sequence converges and its limit is zero.
Now, let’s work on a couple more examples that are designed to make sure that we don’t get too used to relying on our intuition with these problems. As we noted in the previous section our intuition can often lead us astray with some of the concepts we’ll be looking at in this chapter.
Example 2: Determine if the following sequences are monotonic and/or bounded.
We’ll start with the bounded part of this example first and then come back and deal with the increasing/decreasing question since that is where students often make mistakes with this type of sequence.
Now let’s think about the monotonic question.
Before moving on to the next part there is a natural question that many students will have at this point.
Ques: Why did we use Calculus to determine the increasing/decreasing nature of the sequence when we could have just plugged in a couple of n’s and quickly determined the same thing?
The answer to this question is the next part of this example!
This is a messy-looking sequence, but it needs to be in order to make the point of this part.
Note - So, as the last example has shown we need to be careful in making assumptions about sequences. Our intuition will often not be sufficient to get the correct answer and we can NEVER make assumptions about a sequence based on the value of the first few terms. As the last part has shown there are sequences which will increase or decrease for a few terms and then change direction after that.
We’ll close out this section with a nice theorem that we’ll use in some of the proofs later in this chapter.
Theorem
If {an} is bounded and monotonic then {an} is convergent.
Be careful to not misuse this theorem. It does not say that if a sequence is not bounded and/or not monotonic that it is divergent. Example 2b is a good case in point. The sequence in that example was not monotonic but it does converge.
Note as well that we can make several variants of this theorem. If {an} is bounded above and increasing then it converges and likewise if {an} is bounded below and decreasing then it converges.
161 videos|58 docs
|
1. What is a sequence in mathematics? |
2. What are the basics of sequence mathematics? |
3. How can sequences be used in real-life situations? |
4. What are arithmetic and geometric sequences? |
5. What are some common applications of sequence mathematics? |
|
Explore Courses for Mathematics exam
|