Many of you are familiar with geometric series. Here is a little derivation of a common result.

Finite Geometric Series

Let

S1 = 1 + a + a^2 + ... + a^n.

We multiply S1 by ‘a’ then see

a*S1 = a+ a^2 + … + a^(n+1).

Subtract the second equation from the first one.

(1-a)*S1 = 1-a^(n+1).

Therefore

S1 = [1-a^(n+1)]/(1-a).

We are sure of this. This result about finite geometric series is in many textbooks.

The

Wikipedia on this is very good.

The infinite case is another story.

If S2 = 1 + b + b^2 + … is an infinite sum then

S2 converges for -1<b<1.