A finite sequence is exactly what it sounds like - an ordered list of numbers that is finite, meaning it has a stopping value. An extensive, uncountable finite sequence also has a stopping value, but it is generally too cumbersome to be practical.

An infinite sequence is an ordered list of numbers, the count of which has no stopping value. Some infinite sequences converge to a desired value and provide an estimation of that value. The sequence of factorials starts with 0!, 1!, 2!, 3!, 4!,… and evaluates to: 1, 1, 2, 6, 12, … However, if we sum up the inverses of the sequence of factorials we get this:

$\frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + ... \approx e$

where e is the natural root = 2.71828… of natural logarithms. The three dots “…” mean the number has no end.