Sequences

# Sequences

 Definition: A Sequence $\{ a_n \}_{n=1}^{\infty}$ is an ordered list of numbers $a_1, a_2, ...$ where $a_j$ denotes the $j^{\mathrm{th}}$ term of the sequence.

One of the most famous sequences is the Fibonacci Sequence that is recursively defined by $f_{1} = 1$, $f_{2} = 1$, and $f_{n} = f_{n-1} + f_{n-2}$ for $n \in \mathrm{N}$ and $n ≥ 3$. In other words, each number of the Fibonacci sequence is obtained by taking the preceding two terms of the Fibonacci sequence and summing them. The first few terms of the Fibonacci sequence are $\{ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... \}$.

Another example of a sequence is when $a_n = \frac{n}{n+1}$. We can calculate the $n^{\mathrm{th}}$ term of this sequence by plugging in some $n \in \mathbb{N}$, for example, the 4th term of this sequence is $a_4 = \frac{4}{4 + 1} = \frac{4}{5}$, and thus $\left \{ a_n \right \} = \left \{ \frac{n}{n + 1} \right \}_{n=1}^{\infty} = \left \{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, ... \right \}$.

One other great example of a sequence is when $a_n = (-1)^n$ and $\{ (-1)^n \}_{n=1}^{\infty} = \{ -1, 1, -1, 1, -1, ... \}$. This might not be the most interesting sequence as there are only two distinct terms in it, but alas, it definitely is a sequence.

We will now look at some important definitions of sequences, many of which are relatively straightforward.

 Definition: The sequence $\{ a_n \}$ is said to be Bounded between $m$ and $M$ where $m ≤ M$ if $\forall n \in \mathbb{N}$, $m ≤ a_n ≤ M$. The sequence is said to be Bounded Above if $\forall n$, $a_n ≤ M$, and Bounded Below if $\forall n$, $m ≤ a_n$.
 Definition: The sequence $\{ a_n \}$ is said to be Strictly Increasing denoted $\{ a_n \} \uparrow$ if $a_{n+1} > a_{n}$ and for every term in the sequence and is said to be Strictly Decreasing denoted $\{ a_n \} \downarrow$ if $a_{n+1} < a_{n}$ for every term in the sequence. If a sequence is increasing or decreasing, we classify the sequence as Monotonic.
 Definition: The sequence $\{ a_n \}$ is said to be Ultimately Increasing if $a_{n+1} > a_{n}$ whenever $n ≥ N$, and is said to be **Ultimately Decreasing if $a_{n+1} < a_{n}$ whenever $n ≥ N$.

We note that the definitions of a sequence being strictly increasing/decreasing different from a sequence being ultimately increasing/decreasing. A sequence that is increasing/decreasing is increasing/decreasing for ALL terms in the sequence. A sequence that is ultimately increasing/decreasing need not increase/decrease for a finite number of terms, but after some $N^{\mathrm{th}}$ term it must indefinitely increase/decrease.

 Definition: The sequence $\{ a_n \}$ is said to be Positive if $a_n > 0$ for all $n \in N$ and the sequence is said to be Negative if $a_n < 0$.
 Definition: The sequence $\{ a_n \}$ is said to be Ultimately Positive if $a_n > 0$ whenever $n ≥ N$ and the sequence is said to be Ultimately Negative if $a_n < 0$ whenever $n ≥ N$.

Like the distinction we made above regarding increasing/decreasing sequences and ultimately increasing/decreasing sequences, the same applies here. A sequence that is positive/negative is positive/negative for ALL of its terms. A sequence that is ultimately positive/negative need not always be one or the other for a finite number of terms, but after the $N^{\mathrm{th}}$ term, all the terms in the sequence must be positive/negative.

## Example 1

Consider the sequence $\{ \frac{2}{2}, \frac{4}{8}, \frac{8}{26}, \frac{16}{80}, ... \}$. Find a formula for the $n^{\mathrm{th}}$ term of this sequence. Determine whether this set is bounded.

We note that the numerators of this sequence are $2^1, 2^2, 2^3, 2^4, ...$, while the denominators seem to be $3^1 - 1, 3^2 - 1, 3^3 - 1, 3^4 - 1, ...$. Therefore a general formula for the $n^{\mathrm{th}}$ term is:

(1)
\begin{align} a_n = \frac{2^n}{3^n - 1} \end{align}

Since $\forall n \in \mathbb{N}$, $2^n ≤ 3^n - 1$ so the fraction $\frac{2^n}{3^n - 1} ≤ 1$ and thus our sequence is bounded above by $M = 1$, that is $a_n ≤ 1$.

## Example 2

Consider the sequence $\{ \frac{1}{1}, \frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{3}}, \frac{-1}{2}, ... \}$. Find a formula for the $n^{\mathrm{th}}$ term of this sequence.

We note that the numerators of this sequence are $1, -1, 1, -1, ...$, while the denominators seem to be the squareroot of successive natural numbers. Therefore a general formula for the $n^{\mathrm{th}}$ term is:

(2)
\begin{align} a_n = \frac{-(-1)^n}{\sqrt{n}} \end{align}

# Sequences as Functions

We can represent a sequence in a variety of ways such as on a number line. For example, consider the sequence in example 1 plotted on a number line: Alternatively, we could represent the sequence as a function whose domain $A := \{ n : n \in \mathbb{N} \}$. For example, the following graph illustrates example 1: 