The sum of a geometric series

One may be curious to find a sum of a geometric series &sumqk. 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 + ... = limn→∞n,

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

The latter can be easily evaluated. Since

n = 1 + q + q2 + q3 + ... + qn,

qn is found by multiplying that term-by-term by q:

qn = q + q2 + q3 + q4 + ... + qn +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 qn +1 that only appears in the second sum qn. Subtraction then gives

n - qn = 1 - qn +1,

or

n = (1 - qn +1) / (1 - q).

We wish to define ∑ as limn→∞n, i.e.

∑ = limn→∞n = limn→∞(1 - qn +1) / (1 - q).

The latter exists and equals 1 / (1 - q) iff |q| < 1, for then and only then limn→∞qn +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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Copyright © 1996-2018 Alexander Bogomolny

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