We commonly refer to a set of events that occur one after the other as a sequence of events. In mathematics, we use the word sequence to refer to an ordered set of numbers, i.e., a set of numbers that “occur one after the other.”
For instance, the numbers 2, 4, 6, 8, …, form a sequence. The order is important; the first number is 2, the second is 4, etc. It seems natural to seek a formula that describes a given sequence, and often this can be done. For instance, the sequence above could be described by the function , for the values of To find the th term in the sequence, we would compute . This leads us to the following, formal definition of a sequence.
We can plot the terms of a sequence with a scatter plot. The horizontal axis is used for the values of , and the values of the terms are plotted on the vertical axis. To visualize this sequence, see Figure 9.1.5.
A pair of coordinate axes are given, with the horizontal axis labeled (rather than ), and the origin at the bottom-left of the image. The first four terms of the sequence are plotted, at coordinates ,,, and . The four points lie in an inverted U shape.
Note that the range of this sequence is finite, consisting of only the values 3 and 5. This sequence is plotted in Figure 9.1.6.
A pair of coordinate axes are shown, with the horizontal axis labeled , and the origin at the bottom-left of the image. The first four terms of the sequence are plotted, illustrating the oscillatory nature of this sequence: the four points make a sort of zig-zag configuration, with the value going back and forth between 3 and 5.
We gave one extra term to begin to show the pattern of signs is “,,,,,,”, due to the fact that the exponent of is a special quadratic. This sequence is plotted in Figure 9.1.7.
Another set of coordinate axes are shown; in this case, the horizontal axis (labeled ) is about two thirds of the way from the bottom of the image. The first five terms of the sequence are shown. Since the triangular numbers follow an “odd, odd, even, even” pattern, the points plotted for the first two terms have negative value, the next two have positive value, and the fifth term is again negative. As increases, the points get closer and closer to the axis.
We should first note that there is never exactly one function that describes a finite set of numbers as a sequence. There are many sequences that start with 2, then 5, as our first example does. We are looking for a simple formula that describes the terms given, knowing there is possibly more than one answer.
Note how each term is 3 more than the previous one. This implies a linear function would be appropriate: for some appropriate value of . If we were to think in terms of ordered pairs, they would be of the form . So one such ordered pair would be . As we want , we set . Thus .
First notice how the sign changes from term to term. This is most commonly accomplished by multiplying the terms by either or . Using multiplies the odd indexed terms by . Thus the first term would be negative and the second term would be positive. Multiplying by multiplies the even indexed terms by . Thus the second term would be negative and the first term would be positive. As this sequence has negative even indexed terms, we will multiply by .
After this, we might feel a bit stuck as to how to proceed. At this point, we are just looking for a pattern of some sort: what do the numbers 2, 5, 10, 17, etc., have in common? There are many correct answers, but the one that we’ll use here is that each is one more than a perfect square. That is, ,,, etc. Thus our formula is .
One who is familiar with the factorial function will readily recognize these numbers. They are ,,,, etc. Since our sequences start with , we cannot write , for this misses the term. Instead, we shift by 1, and write .
This one may appear difficult, especially as the first two terms are the same, but a little “sleuthing” will help. Notice how the terms in the numerator are always multiples of 5, and the terms in the denominator are always powers of 2. Does something as simple as work?
When , we see that we indeed get as desired. When , we get . Further checking shows that this formula indeed matches the other terms of the sequence.
A common mathematical endeavor is to create a new mathematical object (for instance, a sequence) and then apply previously known mathematics to the new object. We do so here. The fundamental concept of calculus is the limit, so we will investigate what it means to find the limit of a sequence.
Definition9.1.9.Limit of a Sequence, Convergent, Divergent.
Let be a sequence and let be a real number. Given any , if an can be found such that for all , then we say the limit of , as approaches infinity, is , denoted
This definition states, informally, that if the limit of a sequence is , then if you go far enough out along the sequence, all subsequent terms will be really close to . Of course, the terms “far enough” and “really close” are subjective terms, but hopefully the intent is clear.
This definition is reminiscent of the - proofs of Chapter 1. In that chapter we developed other tools to evaluate limits apart from the formal definition; we do so here as well.
This definition states, informally, that if the limit of is , then you can guarantee that the terms of will get arbitrarily large (larger than any value of that you think of), by going out far enough in the sequence.
Theorem 9.1.12 allows us, in certain cases, to apply the tools developed in Chapter 1 to limits of sequences. Note two things not stated by the theorem:
If does not exist, we cannot conclude that does not exist. It may, or may not, exist. For instance, we can define a sequence . Let . Since the cosine function oscillates over the real numbers, the limit does not exist. However, for every positive integer ,, so .
If we cannot find a function whose domain contains the positive real numbers where for all in , we cannot conclude does not exist. It may, or may not, exist.
Using Theorem 1.6.21, we can state that . (We could have also directly applied L’Hospital’s Rule.) Thus the sequence converges, and its limit is 3. A scatter plot of every 5 values of is given in Figure 9.1.14. The values of vary widely near , ranging from about to , but as grows, the values approach 3.
A pair of coordinate axes are shown, with the horizontal axis (labeled ) in the center of the image. The range for is from 0 to 100, and every fourth point in the sequence has been plotted. The scatter plot for the sequence follows the graph of . The graph of this function has a vertical asymptote at . Since this is not an integer value, is defined for each , but we can see that the largest negative value of occurs when (to the left of the asymptote), and the largest positive value of occurs when (to the right of the asymptote). As gets large, the coordinate of each point gets closer to 3.
Figure9.1.14.Scatter plot for the sequence in Item 1
The limit does not exist, as oscillates (and takes on every value in infinitely many times). Thus we cannot apply Theorem 9.1.12. The fact that the cosine function oscillates strongly hints that , when is restricted to , will also oscillate. Figure 9.1.15, where the sequence is plotted, shows that this is true. Because only discrete values of cosine are plotted, it does not bear strong resemblance to the familiar cosine wave. The proof of the following statement is beyond the scope of this text, but it is true: there are infinitely many integers that are arbitrarily (i.e., very) close to an even multiple of , so that . Similarly, there are infinitely many integers that are arbitrarily close to an odd multiple of , so that . As the sequence takes on values near 1 and infinitely many times, we conclude that does not exist.
The scatter plot for this example shows the first 100 terms in the sequence . The plot shows a collection of points whose values seem to be randomly scattered between and .
Figure9.1.15.Scatter plot for the sequence in Item 2
We cannot actually apply Theorem 9.1.12 here, as the function is not well defined. (What does mean? In actuality, there is an answer, but it involves complex analysis, beyond the scope of this text.) Instead, we invoke the definition of the limit of a sequence. By looking at the plot in Figure 9.1.16, we would like to conclude that the sequence converges to . Let be given. We can find a natural number such that . Let , and consider :
(since ) .
We have shown that by picking large enough, we can ensure that is arbitrarily close to our limit, , hence by the definition of the limit of a sequence, we can say .
The scatter plot for the sequence shows a pair of coordinate axes, with the horizontal axis labeled , and positioned in the center of the plot.
When is even, the points follow the graph , and get closer and closer to the axis as increases.
When is odd, the points follow the graph , and approach the axis from below.
Figure9.1.16.Scatter plot for the sequence in Item 3
In the previous example we used the definition of the limit of a sequence to determine the convergence of a sequence as we could not apply Theorem 9.1.12. In general, we like to avoid invoking the definition of a limit, and the following theorem gives us tool that we could use in that example instead.
Since this limit is 0, we can apply Theorem 9.1.17 and state that .
Because of the alternating nature of this sequence (i.e., every other term is multiplied by ), we cannot simply look at the limit . We can try to apply the techniques of Theorem 9.1.17:
.
We have concluded that when we ignore the alternating sign, the sequence approaches 1. This means we cannot apply Theorem 9.1.17; it states the the limit must be 0 in order to conclude anything.
The scatter plot for splits into two parts. When is even, the points follow the graph , and approach the line from above as gets large. When is odd, the points follow the graph , and approach the line from below as gets large.
Figure9.1.19.A plot of a sequence in Example 9.1.18, part 2
Since we know that the signs of the terms alternate and we know that the limit of is 1, we know that as approaches infinity, the terms will alternate between values close to 1 and , meaning the sequence diverges. A plot of this sequence is given in Figure 9.1.19.
Since and , we conclude that . So even though we are adding something to each term of the sequence , we are adding something so small that the final limit is the same as before.
Since and , we conclude that .
Since , we have . It does not matter that we multiply each term by 1000; the sequence still approaches 0. (It just takes longer to get close to 0.)
There is more to learn about sequences than just their limits. We will also study their range and the relationships terms have with the terms that follow. We start with some definitions describing properties of the range.
A sequence is said to be bounded if there exist real numbers and such that for all in . The number is called a lower bound for the sequence, and the number is called an upper bound for the sequence.
It follows from this definition that an unbounded sequence may be bounded above or bounded below; a sequence that is both bounded above and below is simply a bounded sequence.
The terms of this sequence are always positive but are decreasing, so we have for all . Thus this sequence is bounded. Figure 9.1.25.(a) illustrates this.
The terms of this sequence obviously grow without bound. However, it is also true that these terms are all positive, meaning . Thus we can say the sequence is unbounded, but also bounded below. Figure 9.1.25.(b) illustrates this.
The points are plotted, for ranging from 1 to 10. These points follow the graph , begining at , and descending toward the axis as gets large.
We can see that the value of remains positive, but will continue to decrease as increases. This illustrates the fact that the sequence is bounded between 0 and 1.
(a)
This time, the points are plotted, for values of between 1 and 8. These points follow the exponential graph ; we can see that the value will continue to increase without bound, illustrating the fact that this sequence is not bounded.
The previous example produces some interesting concepts. First, we can recognize that the sequence converges to 0. This says, informally, that “most” of the terms of the sequence are “really close” to 0. This implies that the sequence is bounded, using the following logic. First, “most” terms are near 0, so we could find some sort of bound on these terms (using Definition 9.1.9, the bound is ). That leaves a “few” terms that are not near 0 (i.e., a finite number of terms). A finite list of numbers is always bounded.
In Example 9.1.22 we saw the sequence , where it was stated that . (Note that this is simply restating part of Theorem 1.3.17. The limit can also be found using logarithms and L’Hospital’s rule.) Even though it may be difficult to intuitively grasp the behavior of this sequence, we know immediately that it is bounded.
Another interesting concept to come out of Example 9.1.24 again involves the sequence . We stated, without proof, that the terms of the sequence were decreasing. That is, that for all . (This is easy to show. Clearly . Taking reciprocals flips the inequality: . This is the same as .) Sequences that either steadily increase or decrease are important, so we give this property a name.
In each of the following, we will examine . If , we conclude that and hence the sequence is increasing. If , we conclude that and the sequence is decreasing. Of course, a sequence need not be monotonic and perhaps neither of the above will apply.
We also give a scatter plot of each sequence. These are useful as they suggest a pattern of monotonicity, but analytic work should be done to confirm a graphical trend.
for all .
Since for all , we conclude that the sequence is decreasing.
for all .
Since for all , we conclude the sequence is increasing.
We can clearly see in Figure 9.1.(c), where the sequence is plotted, that it is not monotonic. However, it does seem that after the first 4 terms it is decreasing. To understand why, perform the same analysis as done before:
.
We want to know when this is greater than, or less than, 0. The denominator is always positive, therefore we are only concerned with the numerator. For small values of , the numerator is positive. As grows large, the numerator is dominated by , meaning the entire fraction will be negative; i.e., for large enough ,. Using the quadratic formula we can determine that the numerator is negative for . In short, the sequence is simply not monotonic, though it is useful to note that for , the sequence is monotonically decreasing.
Again, the plot in Figure 9.1.(d) shows that the sequence is not monotonic, but it suggests that it is monotonically decreasing after the first term. We perform the usual analysis to confirm this.
When , the above expression is ; for , the above expression is . Thus this sequence is not monotonic, but it is monotonically decreasing after the first term.
The scatter plot for shows a sequence that is decreasing, and bounded below by .
(a)
The points in the scatter plot for this example get larger as increases, illustrating the fact that this is an increasing sequence. In fact, for large we can see that , and the points in the plot do appear to follow close to the line .
(b)
The plot for shows a sequence that is initially increasing, but begins to decrease after .
Although this sequence is not monotonic, it is eventually monotonic, and bounded below, which (as we will soon see) is sufficient for the determination of a limit.
(c)
The plot for shows a sequence that is initially increasing, but begins to decrease after .
Although this sequence is not monotonic, it is eventually monotonic, and bounded below, which (as we will soon see) is sufficient for the determination of a limit.
Knowing that a sequence is monotonic can be useful. Consider, for example, a sequence that is monotonically decreasing and is bounded below. We know the sequence is always getting smaller, but that there is a bound to how small it can become. This is enough to prove that the sequence will converge, as stated in the following theorem.
Consider once again the sequence . It is easy to show it is monotonically decreasing and that it is always positive (i.e., bounded below by 0). Therefore we can conclude by Theorem 9.1.32 that the sequence converges. We already knew this by other means, but in the following section this theorem will become very useful.
We can replace Theorem 9.1.32 with the statement “Let be a bounded, monotonic sequence. Then converges; i.e., exists.” We leave it to the reader in the exercises to show the theorem and the above statement are equivalent.
Sequences are a great source of mathematical inquiry. The On-Line Encyclopedia of Integer Sequences (oeis.org) contains thousands of sequences and their formulae. (As of this writing, there are 328,977 sequences in the database.) Perusing this database quickly demonstrates that a single sequence can represent several different “real life” phenomena.
Interesting as this is, our interest actually lies elsewhere. We are more interested in the sum of a sequence. That is, given a sequence , we are very interested in . Of course, one might immediately counter with “Doesn’t this just add up to ‘infinity’?” Many times, yes, but there are many important cases where the answer is no. This is the topic of series, which we begin to investigate in Section 9.2.
In the following exercises, determine whether the sequence is monotonically increasing or decreasing. If it is not, determine if there is an such that it is monotonic for all .