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 $a(n) = 2n\text{,}$ for the values of $n = 1, 2, \ldots$ To find the 10$^\text{ th }$ term in the sequence, we would compute $a(10)\text{.}$ This leads us to the following, formal definition of a sequence.

##### Definition8.1.1Sequence

A sequence is a function $a(n)$ whose domain is $\mathbb{N}\text{.}$ The range of a sequence is the set of all distinct values of $a(n)\text{.}$

The terms of a sequence are the values $a(1)\text{,}$ $a(2)\text{,}$ …, which are usually denoted with subscripts as $a_1\text{,}$ $a_2\text{,}$ ….

A sequence $a(n)$ is often denoted as $\{a_n\}\text{.}$

Notation: We use $\mathN$ to describe the set of natural numbers, that is, the positive integers $1, 2, 3, \dots$

##### Definition8.1.2

A factorial refers to the product of a descending sequence of natural numbers. For example, the expression $3!$ (read as $3$ factorial) refers to the number $3\cdot2\cdot1 = 6\text{.}$

In general, $n! = n\cdot (n-1)\cdot(n-2)\cdots 2\cdot1\text{,}$ where $n$ is a natural number.

We define $0! = 1\text{.}$ While this does not immediately make sense, it makes many mathematical formulas work properly.

##### Example8.1.3Listing terms of a sequence

List the first four terms of the following sequences.

1. $\{a_n\} = \left\{\frac{3^n}{n!}\right\}$
2. $\{a_n\} = \{4+(-1)^n\}$
3. $\{a_n\} = \left\{\frac{(-1)^{n(n+1)/2}}{n^2}\right\}$

Solution
##### Example8.1.5Determining a formula for a sequence

Find the $n^\text{ th }$ term of the following sequences, i.e., find a function that describes each of the given sequences.

1. $\{a_n\}=\{2, 5, 8, 11, 14, \ldots\}$

2. $\{b_n\}=\{2,-5, 10, -17, 26, -37,\ldots\}$

3. $\{c_n\}=\{1, 1, 2, 6, 24, 120, 720, \ldots\}$

4. $\{d_n\}=\{\frac52, \frac52, \frac{15}8, \frac54,\frac{25}{32},\ldots\}$

Solution

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.

##### Definition8.1.6Limit of a Sequence, Convergent, Divergent

Let $\{a_n\}$ be a sequence and let $L$ be a real number. Given any $\varepsilon>0\text{,}$ if an $N$ can be found such that $\abs{a_n-L}\lt \varepsilon$ for all $n\gt N\text{,}$ then we say the limit of $\{a_n\}\text{,}$ as $n$ approaches infinity, is $L$, denoted \begin{equation*} \lim_{n\to\infty}a_n = L. \end{equation*}

If $\lim\limits_{n\to\infty} a_n$ exists, we say the sequence converges; otherwise, the sequence diverges.

This definition states, informally, that if the limit of a sequence is $L\text{,}$ then if you go far enough out along the sequence, all subsequent terms will be really close to $L\text{.}$ Of course, the terms “far enough” and “really close” are subjective terms, but hopefully the intent is clear.

This definition is reminiscent of the $\varepsilon$–$\delta$ 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.

##### Definition8.1.7Limit of Infinity, Divergent Sequence

Let $\{a_n\}$ be a sequence. We say $\lim\limits_{n\to\infty} a_n=\infty$ if for all $M \gt 0\text{,}$ there exists a number $N$ such that if $n\ge N\text{,}$ then $a_n>M\text{.}$ In this case, we say the sequence diverges to $\infty.$

This definition states, informally, that if the limit of $a_n$ is $\infty\text{,}$ then you can guarantee that the terms of $a_n$ will get arbitrarily large (larger than any value of $M$ that you think of), by going out far enough in the sequence.

Theorem 8.1.8 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:

1. If $\lim\limits_{x\to\infty}f(x)$ does not exist, we cannot conclude that $\lim\limits_{n\to\infty} a_n$ does not exist. It may, or may not, exist. For instance, we can define a sequence $\{a_n\} = \{\cos(2\pi n)\}\text{.}$ Let $f(x) = \cos(2\pi x)\text{.}$ Since the cosine function oscillates over the real numbers, the limit $\lim\limits_{x\to\infty}f(x)$ does not exist. However, for every positive integer $n\text{,}$ $\cos(2\pi n) = 1\text{,}$ so $\lim\limits_{n\to\infty} a_n = 1\text{.}$

2. If we cannot find a function $f(x)$ whose domain contains the positive real numbers where $f(n) = a_n$ for all $n$ in $\mathbb{N}\text{,}$ we cannot conclude $\lim\limits_{n\to\infty} a_n$ does not exist. It may, or may not, exist.

##### Example8.1.9Determining convergence/divergence of a sequence

Determine the convergence or divergence of the following sequences.

1. $\ds\{a_n\} = \left\{\frac{3n^2-2n+1}{n^2-1000}\right\}$
2. $\{b_n\} = \{\cos(n) \}$
3. $\ds\{c_n\} = \left\{\frac{(-1)^n}{n}\right\}$

Solution

It seems that $\{(-1)^n/n\}$ converges to 0 but we lack the formal tool to prove it. The following theorem gives us that tool.

##### Example8.1.12Determining the convergence/divergence of a sequence

Determine the convergence or divergence of the following sequences.

1. $\ds \{a_n\} = \left\{\frac{(-1)^n}{n}\right\}$
2. $\ds \{a_n\} = \left\{\frac{(-1)^n(n+1)}{n}\right\}$

Solution

We continue our study of the limits of sequences by considering some of the properties of these limits.

##### Example8.1.15Applying properties of limits of sequences

Let the following sequences, and their limits, be given:

• $\ds \{a_n\} = \left\{\frac{n+1}{n^2}\right\}\text{,}$ and $\lim\limits_{n\to\infty} a_n = 0\text{;}$

• $\ds \{b_n\} = \left\{\left(1+\frac1n\right)^{n}\right\}\text{,}$ and $\lim\limits_{n\to\infty} b_n = e\text{;}$ and

• $\ds \{c_n\} = \big\{n\cdot \sin(5/n)\big\}\text{,}$ and $\lim\limits_{n\to\infty} c_n = 5\text{.}$

Evaluate the following limits.

1. $\lim\limits_{n\to\infty} (a_n+b_n)$
2. $\lim\limits_{n\to\infty} (b_n\cdot c_n)$
3. $\lim\limits_{n\to\infty} (1000\cdot a_n)$

Solution

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.

##### Definition8.1.16Bounded and Unbounded Sequences

A sequence $\{a_n\}$ is said to be bounded if there exists real numbers $m$ and $M$ such that $m \lt a_n \lt M$ for all $n$ in $\mathbb{N}\text{.}$

A sequence $\{a_n\}$ is said to be unbounded if it is not bounded.

A sequence $\{a_n\}$ is said to be bounded above if there exists an $M$ such that $a_n \lt M$ for all $n$ in $\mathbb{N}\text{;}$ it is bounded below if there exists an $m$ such that $m\lt a_n$ for all $n$ in $\mathbb{N}\text{.}$

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.

##### Example8.1.17Determining boundedness of sequences

Determine the boundedness of the following sequences.

1. $\ds\{a_n\} = \left\{\frac1n\right\}$
2. $\{a_n\} = \{2^n\}$

Solution

The previous example produces some interesting concepts. First, we can recognize that the sequence $\ds\left\{1/n\right\}$ 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 8.1.6, the bound is $\varepsilon$). 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.

This logic implies that if a sequence converges, it must be bounded. This is indeed true, as stated by the following theorem.

Keep in mind what Theorem 8.1.19 does not say. It does not say that bounded sequences must converge, nor does it say that if a sequence does not converge, it is not bounded.

In Example 8.1.15 we saw the sequence $\ds \{b_n\} = \left\{\left(1+1/n\right)^{n}\right\}\text{,}$ where it was stated that $\lim\limits_{n\to\infty} b_n = e\text{.}$ (Note that this is simply restating part of Theorem 1.3.12. The limit can also be found using logarithms and L'Hopital'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 8.1.17 again involves the sequence $\{1/n\}\text{.}$ We stated, without proof, that the terms of the sequence were decreasing. That is, that $a_{n+1} \lt a_n$ for all $n\text{.}$ (This is easy to show. Clearly $n \lt n+1\text{.}$ Taking reciprocals flips the inequality: $1/n > 1/(n+1)\text{.}$ This is the same as $a_n > a_{n+1}\text{.}$) Sequences that either steadily increase or decrease are important, so we give this property a name.

##### Definition8.1.20Monotonic Sequences
1. A sequence $\{a_n\}$ is monotonically increasing if $a_n \leq a_{n+1}$ for all $n\text{,}$ i.e., \begin{equation*} a_1 \leq a_2 \leq a_3 \leq \cdots a_n \leq a_{n+1} \cdots \end{equation*}

2. A sequence $\{a_n\}$ is monotonically decreasing if $a_n \geq a_{n+1}$ for all $n\text{,}$ i.e., \begin{equation*} a_1 \geq a_2 \geq a_3 \geq \cdots a_n \geq a_{n+1} \cdots \end{equation*}

3. A sequence is monotonic if it is monotonically increasing or monotonically decreasing.

It is sometimes useful to call a monotonically increasing sequence strictly increasing if $a_n \lt a_{n+1}$ for all $n\text{;}$ i.e, we remove the possibility that subsequent terms are equal.

A similar statement holds for strictly decreasing.

##### Example8.1.21Determining monotonicity

Determine the monotonicity of the following sequences.

1. $\ds \{a_n\} = \left\{\frac{n+1}n\right\}$

2. $\ds \{a_n\} = \left\{\frac{n^2+1}{n+1}\right\}$

3. $\ds \{a_n\} = \left\{\frac{n^2-9}{n^2-10n+26}\right\}$

4. $\ds \{a_n\} = \left\{\frac{n^2}{n!}\right\}$

Solution

Knowing that a sequence is monotonic can be useful. In particular, if we know that a sequence is bounded and monotonic, we can conclude it converges! 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 $\{a_n\} = \{1/n\}\text{.}$ 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 8.1.26 that the sequence converges. We already knew this by other means, but in the following section this theorem will become very useful.

Sequences are a great source of mathematical inquiry. The On-Line Encyclopedia of Integer Sequences (http://oeis.org) contains thousands of sequences and their formulae. (As of this writing, there are 257,537 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 $\{a_n\}\text{,}$ we are very interested in $a_1+a_2+a_3+\cdots\text{.}$ 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 8.2.

# Subsection8.1.1Exercises

Terms and Concepts

In the following exercises, give the first five terms of the given sequence.