Definition of a Sequence, calculus lesson

A savings account balance checked once a year, the number of bacteria counted every hour, the decimal approximations $1, 1.4, 1.41, 1.414$ of $\sqrt{2}$ . Each is an endless list of numbers in a definite order. That ordered list is a sequence, and before asking where it is heading you need words for how it behaves along the way.

Key terms

Sequence as a function on the natural numbers

Definition A sequence of real numbers is a function $a$ whose domain is the set of positive integers $\{1, 2, 3, \dots\}$ , or more generally a set of integers $\{n_0, n_0 + 1, n_0 + 2, \dots\}$ , and whose values are real numbers. Its value at $n$ is written $a_n$ instead of $a(n)$ , and the whole sequence is written $\{a_n\}$ or $\{a_n\}_{n=1}^{\infty}$ .

In plain words A sequence is a rule that assigns a number to each position $1, 2, 3$ and so on. Unlike a function on an interval, it only has values at whole numbers.

Example The sequence $a_n = \frac{1}{n}$ is the list $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$ . It is the function $f(x) = \frac{1}{x}$ looked at only at $x = 1, 2, 3, \dots$ .

Terms

Definition The number $a_n$ is the $n$ th term of the sequence $\{a_n\}$ , and $n$ is its index. Two sequences $\{a_n\}$ and $\{b_n\}$ with the same set of indices are equal when $a_n = b_n$ for every index $n$ .

In plain words The terms are the entries of the list, and the index says where each one sits.

Example The sequence

a_n = (-1)^n \frac{n}{n + 1}

has first terms $a_1 = -\frac{1}{2}$ , $a_2 = \frac{2}{3}$ , $a_3 = -\frac{3}{4}$ , $a_4 = \frac{4}{5}$ . The sequences $a_n = n$ and $b_n = n^2$ share the terms $1$ and $4$ as values, but they are different sequences since $a_2 = 2 \neq 4 = b_2$ .

Explicit and recursive rules

Definition An explicit rule gives $a_n$ directly as an expression in $n$ . A recursive rule gives one or more starting terms and an equation that expresses $a_{n+1}$ through earlier terms, such as $a_{n+1} = g(a_n)$ for $n \ge 1$ . A recursive rule determines a unique sequence once the starting terms are given.

In plain words With an explicit rule you can jump straight to the hundredth term. With a recursive rule you must walk there step by step, but the rule is often the natural way the sequence arises.

Example The rule $a_1 = 1$ , $a_{n+1} = 2a_n + 1$ gives $1, 3, 7, 15, 31$ . These are one less than powers of $2$ , which suggests the explicit rule $a_n = 2^n - 1$ . It checks out, since $2^1 - 1 = 1$ and $2(2^n - 1) + 1 = 2^{n+1} - 1$ .

Bounded sequences

Definition A sequence $\{a_n\}$ is bounded above if there is a number $M$ with $a_n \le M$ for all $n$ , and bounded below if there is a number $m$ with $a_n \ge m$ for all $n$ . It is bounded if it is both, which is the same as having a number $K$ with $\lvert a_n \rvert \le K$ for all $n$ .

In plain words A bounded sequence stays inside some fixed band. It never runs off to $\infty$ or $-\infty$ .

Example For $a_n = \frac{n}{n + 1}$ we have $\frac{1}{2} \le a_n < 1$ for every $n \ge 1$ , so it is bounded. The sequence $a_n = n^2$ is bounded below by $1$ but not bounded above.

Positive sequences

Definition A sequence is positive if $a_n > 0$ for every $n$ , and nonnegative if $a_n \ge 0$ for every $n$ . It is eventually positive if there is an $N$ with $a_n > 0$ for all $n \ge N$ .

In plain words All the terms are above zero. Many later tests for series only apply to sequences like this, so it is worth checking first.

Example For $a_n = n^2 - 5n + 7$ , the quadratic $x^2 - 5x + 7$ has discriminant $25 - 28 < 0$ and opens upward, so it is positive for every real $x$ . The terms $3, 1, 1, 3, 7$ confirm it. By contrast $b_n = n - 3$ is not positive, but it is eventually positive.

Increasing and decreasing

Definition A sequence $\{a_n\}$ is increasing if $a_n < a_{n+1}$ for all $n$ , and decreasing if $a_n > a_{n+1}$ for all $n$ . With non-strict inequalities, $a_n \le a_{n+1}$ or $a_n \ge a_{n+1}$ , it is called nondecreasing or nonincreasing. Some books use increasing for the non-strict version, so read the definition in use.

In plain words Increasing means each term is bigger than the one before. You can check the sign of $a_{n+1} - a_n$ , compare $a_{n+1}/a_n$ with $1$ when the terms are positive, or, when $a_n = f(n)$ for a function $f$ differentiable on $[1, \infty)$ , show that $f'(x) > 0$ for all $x \ge 1$ .

Example For $a_n = \frac{n}{n + 1}$ ,

a_{n+1} - a_n = \frac{n + 1}{n + 2} - \frac{n}{n + 1} = \frac{(n + 1)^2 - n(n + 2)}{(n + 1)(n + 2)} = \frac{1}{(n + 1)(n + 2)} > 0,

so the sequence is increasing.

Monotonic sequences

Definition A sequence is monotonic if it is nondecreasing or nonincreasing. It is eventually monotonic if this holds for all $n \ge N$ for some $N$ .

In plain words A monotonic sequence never changes direction. It may stand still for a step but never turns back.

Example For $a_n = 2^n / n!$ the terms are positive and

\frac{a_{n+1}}{a_n} = \frac{2^{n+1}}{(n + 1)!} \cdot \frac{n!}{2^n} = \frac{2}{n + 1},

which equals $1$ for $n = 1$ and is less than $1$ for $n \ge 2$ .

$n$	$1$	$2$	$3$	$4$	$5$
$a_n$	$2$	$2$	$\frac{4}{3}$	$\frac{2}{3}$	$\frac{4}{15}$

So the sequence is nonincreasing, hence monotonic, and it is strictly decreasing from $n = 2$ on.

Alternating sequences

Definition A sequence is alternating if consecutive terms have opposite signs, that is, $a_n a_{n+1} < 0$ for all $n$ . Equivalently $a_n = (-1)^n b_n$ for all $n$ or $a_n = (-1)^{n+1} b_n$ for all $n$ , with $b_n > 0$ .

In plain words The signs flip at every step, plus, minus, plus, minus. An alternating sequence is never monotonic.

Example Since $\cos(n\pi) = (-1)^n$ , the sequence

a_n = \frac{\cos(n\pi)}{n} = \frac{(-1)^n}{n}

is $-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, \dots$ . It has the form $(-1)^n b_n$ with $b_n = 1/n > 0$ , so it alternates.

Check yourself

Write the first four terms of $a_1 = 3$ , $a_{n+1} = \frac{a_n}{2} + 1$ , and check that $a_n = 2 + 2^{1-n}$ fits them.

Show the answer

The terms are $3, \frac{5}{2}, \frac{9}{4}, \frac{17}{8}$ . The explicit rule gives $2 + 1$ , $2 + \frac{1}{2}$ , $2 + \frac{1}{4}$ , $2 + \frac{1}{8}$ , the same numbers.

Is $a_n = (n - 3)^2$ monotonic? Is it bounded?

Show the answer

The terms are $4, 1, 0, 1, 4, 9, \dots$ , which go down and then up, so it is not monotonic, though it is increasing from $n = 3$ on. It is bounded below by $0$ but not bounded above, so it is not bounded.

Is $a_n = \frac{(-1)^n}{n}$ bounded? Is it monotonic?

Show the answer

It is bounded, since $\lvert a_n \rvert \le 1$ . It alternates in sign, so it is not monotonic.

Definition of a Sequence

Key terms

Sequence as a function on the natural numbers

Terms

Explicit and recursive rules

Bounded sequences

Positive sequences

Increasing and decreasing

Monotonic sequences

Alternating sequences

Check yourself

How this lesson connects

Used later in