Sequences and Series

If you were to participate in a TV show where the $10,000 question was "Given the numbers 2, 4, 8, 16, and 32, what comes next in the sequence?", what would your best guess be? If your response is 64, then congratulations—you just came closer to understanding one of the key concepts in mathematical abstraction: that of a sequence. A sequence is, pretty much like in the ordinary sense of the word, a particular order in which things follow each other. Here, things are (in most cases) integers or real numbers that are related. The order of the elements matters. The elements are also called the members or terms of the sequence.

For example, in the preceding sequence of the TV show you participated in, every term stems from the number prior being multiplied by 2; there is no end in this sequence as there is no end in the number of terms (integer numbers) you can come up with. In other instances, elements in a sequence can appear more than once. Think...