How do you Use an infinite geometric series to express a repeating decimal as a fraction?

1 Answer
Sep 19, 2014

Let us find a fraction for

0.121212...

by splitting it into

=0.12+0.0012+0.000012+cdots

by rewriting each term as a fraction,

=12/100+12/10000+12/1000000+cdots

by rewriting a little further,

=12/100+12/100(1/100)+12/100(1/100)^2+cdots

The series above is a geometric series with

a=12/100 and r=1/100

Hence, the sum is

0.121212...=a/{1-r}={12/100}/{1-1/100}={12/100}/{99/100}=12/99=4/33