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Mathematics LibreTexts

8.1: Sequences

Skills to Develop

In this section, we strive to understand the ideas generated by the following important questions:

  • What is a sequence?
  • What does it mean for a sequence to converge?
  • What does it mean for a sequence to diverge?

We encounter sequences every day. Your monthly rent payments, the annual interest you earn on investments, a list of your car’s miles per gallon every time you fill up; all are examples of sequences. Other sequences with which you may be familiar include the Fibonacci sequence

\[1, 1, 2, 3, 5, 8, . . . \nonumber\]

in which each entry is the sum of the two preceding entries and the triangular numbers

\[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . . .\nonumber\]

which are numbers that correspond to the number of vertices seen in the triangles in Figure 8.1. Sequences of integers are of such interest to mathematicians and others that 

Figure 8.1: Triangular numbers

they have a journal1 devoted to them and an on-line encyclopedia2 that catalogs a huge number of integer sequences and their connections. Sequences are also used in digital recordings and digital images. To this point, most of our studies in calculus have dealt with continuous information (e.g., continuous functions). The major difference we will see now is that sequences model discrete instead of continuous information. We will study ways to represent and work with discrete information in this chapter as we investigate sequences and series, and ultimately see key connections between the discrete and continuous.

Preview Activity \(\PageIndex{1}\)

Suppose you receive \($5000\) through an inheritance. You decide to invest this money into a fund that pays \(8\%\) annually, compounded monthly. That means that each month your investment earns \(0.08 12 · P\) additional dollars, where \(P\) is your principal balance at the start of the month. So in the first month your investment earns \(5000 \left( \dfrac{0.08}{12} \right)\) or \($33.33\). If you reinvest this money, you will then have \($5033.33\) in your account at the end of the first month. From this point on, assume that you reinvest all of the interest you earn.

  1. How much interest will you earn in the second month? How much money will you have in your account at the end of the second month?
  2. Complete Table 8.1 to determine the interest earned and total amount of money in this investment each month for one year.
  3. As we will see later, the amount of money \(P_n\) in the account after month n is given by \[P_n = 5000 \left( 1 + \dfrac{0.08}{12} \right)^n \)\]Use this formula to check your calculations in Table 8.1. Then find the amount of money in the account after \(5\) years. 

1The Journal of Integer Sequences at http://www.cs.uwaterloo.ca/journals/JIS/

2The On-Line Encyclopedia of Integer Sequences at http://oeis.org/

Month Interest earned Total amount of money in the account
0 $0 $5000.00
1 $33.33 $5033.33
2    
3    
4    
5    
6    
7    
8    
9    
10    
11    
12    

Table 8.1: Interest

d. How many years will it be before the account has doubled in value to $10000?

Sequences

As our discussion in the introduction and Preview Activity \(\PageIndex{1}\) illustrate, many discrete phenomena can be represented as lists of numbers (like the amount of money in an account over a period of months). We call these any such list a sequence. In other words, a sequence is nothing more than list of terms in some order. To be able to refer to a sequence in a general sense, we often list the entries of the sequence with subscripts,

\[s_1, s_2, . . ., s_n . . .,\nonumber\]

where the subscript denotes the position of the entry in the sequence. More formally,

Definition: Sequences

 A sequence is a list of terms \(s_1, s_2, s_3, . . .\) in a specified order.

As an alternative to Definition 8.1, we can also consider a sequence to be a function f whose domain is the set of positive integers. In this context, the sequence \(s_1, s_2, s_3\), . . . would correspond to the function \(f\) satisfying \(f (n) = s_n\) for each positive integer n. This alternative view will be be useful in many situations. We will often write the sequence \(s_1, s_2, s_3, . . .\) using the shorthand notation {\(s_n\)}. The value \(s_n\) (alternatively \(s(n)\)) is called the nth term in the sequence. If the terms are all 0 after some fixed value of \(n\), we say the sequence is finite. Otherwise the sequence is infinite. We will work with both finite and infinite sequences, but focus more on the infinite sequences. With infinite sequences, we are often interested in their end behavior and the idea of convergent sequences.

Activity \(\PageIndex{1}\)

  1. Let \(s_n\) be the nth term in the sequence 1, 2, 3, . . .. Find a formula for \(s_n\) and use appropriate technological tools to draw a graph of entries in this sequence by plotting points of the form \((n, s_n)\) for some values of n. Most graphing calculators can plot sequences; directions follow for the TI-84.
    • In the MODE menu, highlight SEQ in the FUNC line and press ENTER.
    • In the Y= menu, you will now see lines to enter sequences. Enter a value for \(nMin\) (where the sequence starts), a function for \(u(n)\) (the nth term in the sequence), and the value of \(u_n Min\).
    • Set your window coordinates (this involves choosing limits for \(n\) as well as the window coordinates XMin, XMax, YMin, and YMax.
    • The GRAPH key will draw a plot of your sequence. Using your knowledge of limits of continuous functions as \( x \rightarrow \infty \), decide if this sequence {\(s_n\)} has a limit as \(n \rightarrow \infty\). Explain your reasoning.
  2. Let \(s_n\) be the \(n\)th term in the sequence \(1, \frac{1}{2}, \frac{1}{3}, ...\). Find a formula for \(s_n\). Draw a graph of some points in this sequence. Using your knowledge of limits of continuous functions as \( x \rightarrow \infty \), decide if this sequence {\(s_n\)} has a limit as \( n \rightarrow \infty \). Explain your reasoning.
  3. Let \(s_n\) be the \(n\)th term in the sequence \(2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}. . .\). Find a formula for \(s_n\). Using your knowledge of limits of continuous functions as \(x \rightarrow \infty \), decide if this sequence {\(s_n\)} has a limit as \( n \rightarrow \infty \). Explain your reasoning.

Next we formalize the ideas from Activity \(\PageIndex{1}\).

Activity \(\PageIndex{2}\)

  1. Recall our earlier work with limits involving infinity in Section 2.8. State clearly what it means for a continuous function \(f\) to have a limit \(L\) as \( x \rightarrow \infty \).
  2. Given that an infinite sequence of real numbers is a function from the integers to the real numbers, apply the idea from part (a) to explain what you think it means for a sequence {\(s_n\)} to have a limit as \( n \rightarrow \infty \).
  3. Based on your response to (b), decide if the sequence { \(\frac{1+n}{2+n}\) } has a limit as \( n \rightarrow \infty \). If so, what is the limit? If not, why not? 

In Activities \(\PageIndex{1}\) and \(\PageIndex{2}\) we investigated the notion of a sequence {\(s_n\)} having a limit as \(n\) goes to infinity. If a sequence {\(s_n\)} has a limit as \(n\) goes to infinity, we say that the sequence converges or is a convergent sequence. If the limit of a convergent sequence is the number \(L\), we use the same notation as we did for continuous functions and write

\[ \lim_{n \rightarrow \infty} s_n = L. \nonumber\]

If a sequence {\(s_n\)} does not converge then we say that the sequence {\(s_n\)} diverges. Convergence of sequences is a major idea in this section and we describe it more formally as follows.

Definition: Convergence

A sequence {\(s_n\)} of real numbers converges to a number \(L\) if we can make all values of \(s_k\) for \(k \geq n\) as close to \(L\) as we want by choosing \(n\) to be sufficiently large.

Remember, the idea of sequence having a limit as \( n \rightarrow \infty \) is the same as the idea of a continuous function having a limit as \( x \rightarrow \infty \). The only new wrinkle here is that our sequences are discrete instead of continuous. We conclude this section with a few more examples in the following activity.  

Activity \(\PageIndex{3}\)

Use graphical and/or algebraic methods to determine whether each of the following sequences converges or diverges.

  1. {\(\frac{1+2n}{3n−2}\)}
  2. { \(\frac{5+3^n}{10+2^n} \) }
  3. {\(\frac{10^n}{n!} \)} (where ! is the factorial symbol and \(n! = n(n − 1)(n − 2) · · · (2)(1)\) for any positive integer \(n\) (as convention we define \(0!\) to be \(1\))).

Summary

In this section, we encountered the following important ideas:

  • A sequence is a list of objects in a specified order. We will typically work with sequences of real numbers and can also think of a sequence as a function from the positive integers to the set of real numbers.
  • A sequence {\(s_n\)} of real numbers converges to a number \(L\) if we can make every value of \(s_k\) for \(k \geq n\) as close as we want to \(L\) by choosing \(n\) sufficiently large.
  • A sequence diverges if it does not converge.

Contributors

Matt Boelkins (Grand Valley State University), David Austin (Grand Valley State University), Steve Schlicker (Grand Valley State University)