# Sequence

## Definition

A sequence in a set $S$ is a function from the set of natural numbers $\mathbb{N} = \{ 1,2,3,\dots \}$ to $S$.

The way such a sequence is described is simply by listing the images of 1,2,3,... in the right order. Explicitly, for a function $f: \mathbb{N} \to S$, the sequence can be written as:

$f(1), f(2), f(3), f(4), f(5), \dots$

The values $f(n), n \in \mathbb{N}$ are called the terms of the sequence. Specifically, the value $f(n)$ is called the $n^{th}$ term.

For instance, the sequence given by the function $f(n) := n^2$ can be written as:

$1, 4, 9, 16, 25, 36, 49, 64, \dots$

Note that it is not possible to unambiguously describe a sequence (which is infinite) by just listing the first few terms (of which there are only finitely many), but the general idea behind listing the first few terms and putting the ellipses ("...") is that people are expected to figure out the most natural choice of function that fits the first few terms.

## Notation

Instead of using the typical function notation with the input to the function in parentheses, sequences are typically notated using a subscript notation. The sequence is named by a letter, and individual terms of the sequence are denoted by that letter with a subscript used for the position (index). For instance, for a sequence denoted with letter $a$, the first term is denoted $a_1$, the second term is denoted $a_2$, and the $n^{th}$ term is denoted $a_n$. The sequence itself is written with the shorthand $(a_n)_{n \in \mathbb{N}}$.

## Terminology

Term Meaning In subscript notation for a sequence $(a_n)_{n \in \mathbb{N}}$ In function notation for sequence given by a function $n$
term of a sequence the appropriate function value $a_n$ is the $n^{th}$ term $f(n)$ is the $n^{th}$ term
index or position of a term the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. the index of $a_n$ is $n$. the index of $f(n)$ is $n$.
successor or "next term" the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the $n^{th}$ term is the $(n + 1)^{th}$ term.
Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.
The successor to $a_n$ is $a_{n+1}$ The successor to $f(n)$ is $f(n + 1)$
predecessor or "previous term" the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the $n^{th}$ term is the $(n - 1)^{th}$ term.<
Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.
The first term doesn't have a predecessor.
The predecessor to $a_n$ is $a_{n-1}$ if $n > 1$. The predecessor to $f(n)$ is $f(n - 1)$.
repetition-free sequence a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. If $m,n \in \mathbb{N}$ with $m \ne n$, then $a_m \ne a_n$. If $m,n \in \mathbb{N}$ with $m \ne n$, then $f(m) \ne f(n)$.
constant sequence a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other. For all $m,n \in \mathbb{N}$, $a_m = a_n$ For all $m,n \in \mathbb{N}$, $f(m) = f(n)$.
eventually constant sequence a sequence for which there exists a natural number $n_0$ such that the part of the sequence beyond that point is constant For $m,n \in \mathbb{N}$ with $m,n \ge n_0$, $a_m = a_n$ For $m,n \in \mathbb{N}$ with $m,n \ge n_0$, $f(m) = f(n)$
periodic sequence a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number $h$ such that for all natural numbers $n$, the $n^{th}$ term equals the $(n + h)^{th}$ term. The smallest such $h$ is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1. $a_n = a_{n + h} \ \forall \ n \in \mathbb{N}$ $f(n) = f(n + h) \ \forall \ n \in \mathbb{N}$
eventually periodic sequence a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers $n_0$ and $h$ such that for all natural numbers $n \ge n_0$, the $n^{th}$ term equals the $(n + h)^{th}$ term. $a_n = a_{n + h}$ for all $n \in \mathbb{N}$ satisfying $n \ge n_0$ $f(n) = f(n + h)$ for all $n \in \mathbb{N}$ satisfying $n \ge n_0$
range of a sequence the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. The set $\{ a_n \mid n \in \mathbb{N} \}$ The set $\{ f(n) \mid n \in \mathbb{N} \}$

There are various other notions associated with sequences specifically in the context of sequences that take values in the real numbers, i.e., sequences of real numbers.

Term Meaning
increasing sequence (sometimes called strictly increasing sequence to distinguish it from non-decreasing sequence) If $m < n$, the $m^{th}$ term is less than the $n^{th}$ term. Note that it suffices to check that each term is less than the next term.
non-decreasing sequence (sometimes called weakly increasing sequence or monotonically increasing sequence) If $m < n$, the $m^{th}$ term is less than or equal to the $n^{th}$ term. Note that it suffices to check that each term is less than or equal to the next term.
decreasing sequence (sometimes called strictly decreasing sequence to distinguish it from non-increasing sequence) If $m < n$, the $m^{th}$ term is greater than the $n^{th}$ term. Note that it suffices to check that each term is less than the next term.
non-increasing sequence (sometimes called weakly decreasing sequence or monotonically decreasing sequence) If $m < n$, the $m^{th}$ term is greater than or equal to the $n^{th}$ term. Note that it suffices to check that each term is less than or equal to the next term.
monotone sequence Either non-decreasing or non-increasing
bounded sequence The range of the sequence is a bounded subset of the reals (here, bounded means bounded both from above and from below). Note that any constant or periodic sequence is bounded. Also, any convergent sequence (defined later) is bounded.
sequence bounded from above The range of the sequence is bounded from above in the reals. Note that any bounded sequence is bounded from above. Also, any non-increasing sequence is bounded from above.
sequence bounded from below The range of the sequence is bounded from above in the reals. Note that any bounded sequence is bounded from below. Also, any non-decreasing sequence is bounded from below.
convergent sequence a sequence that has a limit.

Note that the notions of bounded and convergent are special in that both notions refer to eventual behavior only. A sequence is bounded if and only if it is eventually bounded, i.e., the first few finitely many terms do not affect whether or not the sequence is bounded. Similarly, a sequence is convergent if and only if it is eventually convergent.

## Sequences indexed from zero onward

For a number of applications, particularly power series, it is useful to consider sequences that are indexed starting from zero. Such sequences can be thought of as functions from the set $\mathbb{N}_0 = \mathbb{N} \cup \{ 0 \} = \{ 0,1,2,\dots \}$. All the definitions and concepts developed for sequences can be considered for sequences indexed from zero onward.

## Operations on sequences

### Pointwise operations

We can do various pointwise operations on sequences just as we do other types of pointwise combination of functions, such as a pointwise sum, difference, product, or quotient (assuming the second sequence has no zero term). Explicitly, for sequences $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$:

• The sum is the sequence whose $n^{th}$ term is $a_n + b_n$
• For a real number $\lambda$, $\lambda(a_n)$ is the sequence whose $n^{th}$ term is $\lambda a_n$.
• The difference is the sequence whose $n^{th}$ term is $a_n - b_n$
• The product is the sequence whose $n^{th}$ term is $a_nb_n$
• The quotient is the sequence whose $n^{th}$ term is $a_n/b_n$. Note that this sequence makes sense only if none of the $b_n$s are zero.

### Shift operations

A shift operation on a sequence takes its terms and moves them all to the left or right.

• A left shift operation moves all the terms a certain amount to the left. The original first few terms disappear. Explicitly, left shifting a sequence $(a_n)$ by a natural number $h$ means the $n^{th}$ term of the new sequence is $a_{n + h}$. The original first $h$ terms have disappeared in the process.
• A right shift operation moves all the terms a certain amount to the right. This introduces the problem that the first few terms of the new sequence are undefined, and need to be specified separately. Explicitly, left shifting a sequence $(a_n)$ by a natural number $h$ means the $n^{th}$ term of the new sequence is $a_{n - h}$. The first $h$ terms of the new sequence need to be specified separately.