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#