A sequence is an infinitely long list of numbers.

Take a look at this **sequence** of fractions:

Notice how the sequence is infinitely long (you can keep scrolling forever) and each term of the sequence is *labeled* with a number.

These numbers, starting with 1 and counting up, are called the “natural numbers”, sometimes indicated by the symbol $\mathbb{N}$:

$\mathbb{N} = \{ \textcolor{#1d4ed8}{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...} \}$Since each natural number corresponds with a term in the pizza sequence, we say that this sequence is *indexed by* the natural numbers.

It's also possible to have a sequence that is indexed by some other set, like the integers: $\mathbb{Z} = \{ ..., -2, -1, 0, 1, 2,... \}$. That kind of sequence would be infinite in both directions, not just to the right.

A sequence is very similar to a function. For example, our pizza sequence is a little bit like the function $f(\textcolor{#1d4ed8}{n}) = \frac{n}{n + 1}$. (Because$f(\textcolor{#1d4ed8}{1}) = \frac 12$, $f(\textcolor{#1d4ed8}{2}) = \frac 23$, and so on.)

But we don't generally use function notation for sequences. Instead, we write them like this:$\left(\frac{n}{n+1}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$This notation basically means "compute $\frac{n}{n + 1}$ for all $n$ in $\mathbb{N}$". Which makes sense, because that's exactly what our pizza sequence is.

But sometimes we don't know exactly what our sequence is. When writing proofs, we will often discuss *mystery sequences*, where we know some *properties* of the sequence but not its exact values. In that case, we can't write an exact formula for the terms, so we just write$(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$instead. Here, $x_n$ is a mystery function that has an unknown value for each $\textcolor{#1d4ed8}{n}$.

If you're in the mood for using mystery notation, you could describe the pizza sequence as "$(x_n)_{n \in \mathbb{N}}$ where $x_n = \frac{n}{n+1}$ for all $n$ in $\mathbb{N}$". Of course, describing the values of $x_n$ really ruins the mystery.

It is much easier to understand the behavior of a sequence if you have a graph:

$\left( \frac{n}{n+1} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

It looks like the terms of this sequence get closer and closer to 1 as $\textcolor{#1d4ed8}{n}$ gets larger. This concept of *converging to a value* is essential in analysis, so keep your eyes peeled for graphs that look like this.

It makes sense that the sequence converges to 1, because for large $\textcolor{#1d4ed8}{n}$, the terms become fractions like $\frac{999,999}{1,000,000}$, which is pretty much 1.

The best way to become familiar with sequences is to look at a lot of them. The following gallery contains a selection of interesting sequences:

$\left( n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

$\left( \frac{1}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

$\left( (-1)^n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

$\left( \frac{(-1)^n}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}$

If you had to classify these sequences into categories, how might you divide them up? (In the next section, we will discuss one way that mathematicians classify sequences.)

Of course! I can't show you all these examples without giving you the chance to build a sequence of your own.

Enter an expression using $\textcolor{#1d4ed8}{n}$ into the box below to create a sequence.