Question

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

Answer 1

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#