Partial Sum of Geometric Series

Theorem

The sum of the first n terms of a geometric series given by initial term a and common ratio r is

a(1−rn)1−r.
Proof

Consider the sum

∑k=0n−1ark=a+ar+⋯+arn−1.

Finding a general expression for the limit of such a sum is just an exercise in re-indexing sums. We proceed as follows

S=∑k=0n−1ark=a∑k=0n−1rk=a+a∑k=1n−1rk=a+ar∑k=1n−1rk−1=a+ar∑k=0n−2rk=a+ar((∑k=0n−2rk)+rn−1−rn−1)=a+ar(∑k=0n−1rk)−arn=a+r(∑k=0n−1ark)−arn=a+rS−arn(1−r)S=a−arnS=a(1−rn)1−r.