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Humans have been trying to get a grasp on infinity since ancient times. In 1948, the computer scientist Alan Turing wrote about a machine with “an unlimited memory capacity obtained in the form of an infinite tape marked out into squares….” Despite the endless nature of the theoretical machine, it would be operated by a finite table of instructions.

To try to understand something about the elusive concept of infinity, mathematicians use various forms of language and symbolism. For instance, an infinite sequence of numbers may be represented this way:

{a_{1}, a_{2}, a_{3}, … a_{n}, a_{(n+1)}, …}

In this case, {a_{1}} would be called the first term, {a_{2}} would be called the second term, and so on. The variable *n* could be any number. The ellipsis {…} indicates no end or limit. Using such terminology expresses a notation for infinity – even if humans do not have a full understanding.

Two types of infinite sequence deserve attention here. An arithmetic infinite sequence is a progression of numbers where the difference between each consecutive term is constant. The interval between the terms is called the “common difference.” For instance, an arithmetic infinite sequence starting with 2 with a common difference of 2 would look like this:

{2, 4, 6, 8, 10 …}

The progression of a geometric infinite sequence is marked by the “common ratio.” For example, a common ratio may indicate that each consecutive number is multiplied by 2. A geometric infinite sequence starting with 2 with a common ratio of x2 would look like this:

{2, 4, 8, 16, 32 …}

The math gets more complex from there. Another form of notation that is used with sequences is called summation or sigma notation. It uses the Greek symbol for the letter sigma.

An infinite sequence should not be confused with an infinite series, which involves adding the numbers instead of listing them.

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