The sum of a geometric series

One may be curious to find a sum of a geometric series &sumq^k. Does any such exist? 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,

∑ = 1 + q + q² + q³ + ... = lim_n→∞∑_n,

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

∑_n = 1 + q + q² + q³ + ... + qⁿ,

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

q ∑_n = q + q² + q³ + q⁴ + ... + q^{n +1},

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

∑_n - q ∑_n = 1 - q^{n +1},

∑_n = (1 - q^{n +1}) / (1 - q).

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

∑ = lim_n→∞∑_n = lim_n→∞(1 - q^{n +1}) / (1 - q).

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,

a + aq + aq² + aq³ + ... = a / (1 - q).

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