An infinite sequence is an infinite list of numbers, often denoted:
$$ a_1, a_2, a_3, ..., a_n, ... $$
where each $a_n$ is any real number.
(Note: the first term of an infinite sequence doesn’t have to be $a_1$; the subscript could be any integer like $a_0$ or $a_{-2419}$.)
Sometimes we can define a sequence based on a closed formula for each term. For example, if we say that $a_n =n$ (i.e. the $n$th term of the sequence is equal to $n)$, then we get this sequence:
$$ 1, 2, 3, ..., n, ... $$
Here’s another example of an infinite sequence:
$$ 0^2, 1^2, 2^2, ..., n^2, ... $$
This sequence can either be written as $a_n = n^2$ for $n \ge 0$ or $a_n = (n-1)^2$ for $n\ge 1$. (Remember, the first term doesn’t have to be $a_1$.)
The closed formula for $a_n$ might not be immediately obvious. For example, consider this sequence:
$$ 1, 1+2, 1+2+3, 1+2+3+4, ... $$
Here, $a_n = 1+2+\cdots+n$. However, there is actually a closed formula for $a_n$:
$$ a_n = \frac{n(n+1)}{2} $$
Not all sequences will have a closed formula for $a_n$. For example, the sequence of harmonic numbers $a_n = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ doesn’t have a closed formula.
Often, we want to know what the terms of a sequence approach, i.e. the value of $\displaystyle\lim_{n\to\infty}a_n$.
Examples: