Geometric Series Part 2

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,
A - r 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, 1 + 0.1 + 0.01 + 0.001 + ... = 1 / (1 - 1/10 ) = 10 / 9.
If r = 1/2, 1 + 1/2 + 1/4 + 1/8 + ... = 1 / ( 1 - 1/2 ) = 2.

If you are interested in more proofs of the geometric series sum, please see these references.

Next: The Final Answer: How Long does the Ball Bounce?

Previous: Lowdown on Geometric Series Part 1

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