The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Bouncing Balls and Geometric Series, Robert Styer and Morgan Besson

## Geometric Series II

We can also find the sum of an infinite geometric series using classical high school algebra.

Given an infinite repeating decimal such as 0.33333..., there is a standard technique to convert it to a fraction:

A = 0.333333...
10 A = 3.33333...

Subtracting the first equation from the second, we get 9 A = 3 and from this we see that A = 3 / 9 = 1 / 3.

A repeating decimal is a disguised form of an infinite geometric series, so it is no accident that we can use this same idea to calculate infinite geometric series!

A = 1 + r + r2 + r3 + ...
r A = r + r2 + r3 + r4 + ...

Subtracting, Ar A = 1 and so A (1− r ) = 1 and so A = 1 / (1 − r ).

In other words,

1 + r + r2 + r3 + r4 + ··· = 1 / (1 − r ).

For instance, if r = 1 / 10, we get

1 + 0.1 + 0.01 + 0.001 + ··· = 1 / (1 − 1 / 10 ) = 10 / 9

If r = 1 / 2, we get

1 + 1 / 2 + 1 / 4 + 1 / 8 + ··· = 1 / ( 1 − 1 / 2 ) = 2

Can you see why our derivation fails if r ≥ 1 or if r ≤ − 1? If you are interested in more proofs of the geometric series sum, please see the annotated bibliography.