Calculus: What Is an Infinite Sequence?

An infinite sequence is essentially an infinite list of numbers. We usually describe an infinite sequence in one of the following ways:

(1) Listing the first few terms, like this: $3, 5, 7, 9, \ldots$

We can do this whenever the sequence follows a clear pattern, so that the reader can tell what the remaining terms will be. The dots indicate that the sequence goes on forever.

(2) Writing a formula for the $n$th term, like this: $x_{n} = n^{2} + 3n$. This is often called a closed form.

We read $x_{n}$ as "x sub n," and it means the $n$th term. This way of describing a sequence is convenient because it allows us to quickly find, say, the 10th term, or the 20th term.

(3) Writing a recursive formula, like this: $a_{n} = a_{n - 1} + 2n$. If we do this, we also need to specify one term, like this: $a_{1} = 0$.

This formula tells us a rule for getting from one term to the next. For example, if we know $a_{3} = 10$, and we want to get $a_{4}$, we apply the formula, where $n = 4$. The formula says that $a_{4} = a_{3} + 2\cdot 4$, so $a_{4} =18$.

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Notice that in (2) we used $x_{n}$ to represent the $n$th term of our sequence, and in (3) we used $a_{n}$. We can use any letter, but $a$ and $x$ are particularly common for sequences.

Now try a problem:

A infinite sequence is just an infinite, ordered list of numbers.

We usually use the notation of the form $a_n$ to denote a sequence. In particular, to define a sequence, we may write any of

Let $\{a_n\}$ be a sequence defined by...

Let $a_n$ be a sequence defined by...

Let $a_1,a_2,\dots$ be a sequence defined by... and many more variations.

$n$ is just a variable that we call the index. In the same way you would say "let $f(x)$ be a function of variable $x$," we have that $a_n$ is a sequence where the index is $n.$

The indexing notation works just how you would expect.

Let $a_1,a_2,\dots,$ denote the sequence of odd integers $1,3,5,\dots$

Ok, you seem to have gotten the hang of this!

Above, we defined infinite sequence using plain English. We also could have defined it in terms of algebra.

Here's some examples of special sequences