8.1: Sequences

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Motivating 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 utility payments, the annual interest you earn on investments, the amount you spend on groceries each week; all are examples of sequences. Other sequences with which you may be familiar include the Fibonacci sequence $1, 1, 2, 3, 5, 8, \ldots \text{,}$ where each term is the sum of the two preceding terms, and the triangular numbers $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots \text{,}$ the number of vertices in the triangles shown in Figure $\PageIndex{1}$.

$8_1_Triangular_Numbers.svg$

Figure $\PageIndex{1}$. Triangular numbers

Sequences of integers are of such interest to mathematicians and others that 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 images.

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

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

Our studies in calculus have dealt with continuous functions. 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 $\frac{0.08}{12} \cdot P$ additional dollars, where $P$ is your principal balance at the start of the month. So in the first month your investment earns

\[ 5000 \left(\frac{0.08}{12}\right) \nonumber \]

or $$33.33\text{.}$ If you reinvest this money, you will then have $$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.

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?

Complete Table $\PageIndex{2}$ to determine the interest earned and total amount of money in this investment each month for one year.

Table $\PageIndex{1}$. Interest
Month	Interest earned	Total amount of money in the account
$0$	$$0.00$	$$5000.00$
$1$	$$33.33$	$$5033.33$
$2$
$3$
$4$
$5$
$6$
$7$
$8$
$9$
$10$
$11$
$12$

As we will see later, the amount of money $P_n$ in the account after month $n$ is given by
\[ P_n = 5000\left(1+\frac{0.08}{12}\right)^{n}\text{.} \nonumber \]

Use this formula to check your calculations in Table $\PageIndex{2}$. Then find the amount of money in the account after 5 years.
How many years will it be before the account has doubled in value to $10000?

Sequences

As Preview Activity $\PageIndex{1}$ illustrates, 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 any such list a sequence. A sequence is nothing more than list of terms in some order. We often list the entries of the sequence with subscripts,

\[ s_1, s_2, \ldots, s_n \ldots\text{,} \nonumber \]

where the subscript denotes the position of the entry in the sequence.

Definition $\PageIndex{1}$

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

We can think of a sequence as a function $f$ whose domain is the set of positive integers where $f(n) = s_n$ for each positive integer $n\text{.}$ This alternative view will be be useful in many situations.

We often denote the sequence

\[ s_1, s_2, s_3, \ldots \nonumber \]

by $\{s_n\}\text{.}$ The value $s_n$ (alternatively $s(n)$) is called the $n$th term in the sequence. If the terms are all 0 after some fixed value of $n\text{,}$ we say the sequence is finite. Otherwise the sequence is infinite. With infinite sequences, we are often interested in their end behavior and the idea of convergent sequences.

Activity $\PageIndex{2}$

Let $s_n$ be the $n$th term in the sequence $1, 2, 3, \ldots\text{.}$ 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 $n\text{.}$ Most graphing calculators can plot sequences; directions follow for the TI-84.
- In the MODEmenu, highlight SEQin the FUNCline 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 $n$th term in the sequence), and the value of u(nMin).
- Set your window coordinates (this involves choosing limits for $n$ as well as the window coordinates XMin, XMax, YMin, and YMax.
- The GRAPHkey will draw a plot of your sequence.
Using your knowledge of limits of continuous functions as $x \to \infty\text{,}$ decide if this sequence $\{s_n\}$ has a limit as $n \to \infty\text{.}$ Explain your reasoning.
Let $s_n$ be the $n$th term in the sequence $1, \frac{1}{2}, \frac{1}{3}, \ldots\text{.}$ Find a formula for $s_n\text{.}$ Draw a graph of some points in this sequence. Using your knowledge of limits of continuous functions as $x \to \infty\text{,}$ decide if this sequence $\{s_n\}$ has a limit as $n \to \infty\text{.}$ Explain your reasoning.
Let $s_n$ be the $n$th term in the sequence $2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots\text{.}$ Find a formula for $s_n\text{.}$ Using your knowledge of limits of continuous functions as $x \to \infty\text{,}$ decide if this sequence $\{s_n\}$ has a limit as $n \to \infty\text{.}$ Explain your reasoning.

Next we formalize the ideas from Activity $\PageIndex{2}$.

Activity $\PageIndex{3}$

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 $x \to \infty\text{.}$
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 $n \to \infty\text{.}$
Based on your response to the part (b), decide if the sequence $\left\{ \frac{1+n}{2+n}\right\}$ has a limit as $n \to \infty\text{.}$ If so, what is the limit? If not, why not?

In Activities $\PageIndex{2}$ and $\PageIndex{3}$ we investigated a sequence $\{s_n\}$ that has a limit as $n$ goes to infinity. More formally, we make the following definition.

Definition $\PageIndex{2}$

A sequence $\{ s_n \}$ converges or is a convergent sequence provided that there is a number $L$ such that we can make $s_n$ as close to $L$ as we want by taking $n$ sufficiently large. In this situation, we call $L$ the limit of the convergent sequence and write

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

If the sequence $\{s_n\}$ does not converge, we say that the sequence $\{s_n\}$ diverges.

The idea of sequence having a limit as $n \to \infty$ is the same as the idea of a continuous function having a limit as $x \to \infty\text{.}$ The only difference is that sequences are discrete instead of continuous.

Activity $\PageIndex{4}$

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

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

Summary

A sequence is a list of objects in a specified order. We will typically work with sequences of real numbers. We can 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 $k \ge n$ as close as we want to $L$ by choosing $n$ sufficiently large.
A sequence diverges if it does not converge.

Month	Interest earned	Total amount of money in the account
\(0\)	\($0.00\)	\($5000.00\)
\(1\)	\($33.33\)	\($5033.33\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(9\)
\(10\)
\(11\)
\(12\)

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