One may be curious to find a sum of a geometric series 1, q, q², q³,... That is, is it possible to associate a number with the expression 1 + q + q² + q³ + ... And, if so, what it might be?

The answer depends on q. For |q| < 1, such a number exists, for other values of q it does not.

The only way to define such a sum is by appealing to the theory of limits. By definition,

where ∑_n is the partial sum of all the terms from the first and up to the n^th which is qⁿ.

The latter can be easily evaluated. Since

q ∑_n is found by multiplying that term-by-term by q:

Note that the two sums share all the terms except for the first (1) that appears only in the first sum ∑_n and the term q^{n +1} that only appears in the second sum q ∑_n. Subtraction then gives

or

We wish to define ∑ as lim_n→∞∑_n, i.e.

The latter exists and equals 1 / (1 - q) iff |q| < 1, for then and only then lim_n→∞q^{n +1} exists and when it does it equals 0.

For a more general series that starts with an arbitrary term a,

Copyright © 1996-2009 Alexander Bogomolny