Introduction to Sequences - At A Glance:
Sequence is a big word. It can sound scary, like an impending thunderstorm heading your direction, approaching with thunderous laughs. Sometimes you could swear you saw a set of evil eyes looking ominously at you from the clouds.
Don't worry. A sequence is just a list of numbers. For example, pick an integer and right it down. Now pick two more and write them down behind the first. Separate each number with a comma. You just made a sequence. Nothing scary about that, right? The individual numbers you just wrote are called terms. The sequence you just made up should have three terms. Because it has a three terms–the list of numbers ends–it is finite.
Let's make it a little more complicated. Get out your writing hand of non-stop, legible scribble and your infinitely long #2 pencil. No, this is not a never-ending standardized exam. Start writing down numbers, separated by a comma, and never stop. That sequence is infinite.
Be Careful: In math, sequences and series are not the same thing.
We can write down a finite sequence by writing down its terms with commas in between. If a sequence is infinite, we don't need a special hand or #2 pencil. Instead, we write down some of its terms, and then we write some dots (...) to indicate that the sequence continues forever in the same pattern.
The following are finite sequences:
1, 2, 3
1, 1, 1, 1, 1, 1, 1, 1
The following are infinite sequences:
1, 1, 1,...
1, 2, 3,....
While you might only encounter infinite sequences in calculus class, nothing about the definition says sequences have to be infinite. Finite sequences are okay, too.