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

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Answer 1 · 2014-09-19T16:27:23

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

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