  ## Infinite Series An infinite series is a series which is infinite.  Duh.  Thus follows the logical mathematical mind in the tradition of the Vulcan’s Doctor Spock.  They’re written in a form such as:

S n  =  0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + ...

where S is the Greek letter, Sigma, and stands as an operator which sums all numbers from n=0 to n=¥ (where the latter symbol, ¥, stands for infinity).  (The “...” stands for “and so forth, ad infinitum.”)

Strictly speaking, the summation can be from zero, one, or other numbers (including negative numbers, up to and including minus infinity), but to qualify as an “infinite” series, there has to be an infinite number of terms.  (One can, however, have a finite series with a finite number of terms.  One such example is given in More Math where the summation is from 0 to 2 (i.e. 0, 1, 2) in one case, and from 0 to 3 in another.)

Examples of a slightly fancier version of summations might include:

S (3 n + 7)  =  7 + 10 + 13 + 16 + 19 + 21 + 24 + ...

or

S an  =  a0 + a1 + a2 + a3 + a4 + a5 + ...

or

S nn  =  0 + 1 + 4 + 27 + 256 + 3,125 + ...

or

S (2n + 1) (-1)n  =  1 -3 + 5 - 7 + 9 - 11 + 13 - ...

One of the criteria for infinite series is whether they “blow up” (i.e. the sum becomes ever larger, going to infinity -- as the first and third clearly do), or whether the series is “well behaved” (i.e. the sum approaches a finite number).  A good example of the latter is:

S (1/2)n   =  1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...  »  2

This infinite series is said to “approach” 2 (as opposed to equal it -- even though it does).

Forward to:

  From the original Star Trek series, with actor Leonard Nimoy playing the part of the pointy eared, extremist in logic.

## The Library of ialexandriah 