Repeating Decimals as Geometric Series

Date: 02/13/2001 at 08:04:14
From: Stephanie
Subject: Geometric series with decimals

I have no idea how to do this. What are the geometric series of 
.666... and of 3.5353535...? Could you please help?

Thank you,
Stephanie

Date: 02/13/2001 at 13:47:39
From: Doctor Greenie
Subject: Re: Geometric series with decimals

Hi, Stephanie -

I'm glad to see you are given these as problems with geometric 
series. Too often, students are taught how to convert repeating 
decimals to common fractions and then later are taught how to find 
the sum of infinite geometric series, without being shown the 
relation between the two processes.

Let's do a couple of problems similar to yours using both methods. I 
will choose the two decimals

     0.27272727...
     4.16666666...

Converting the first decimal to a common fraction by the first 
method, we have

        x =   0.27272727...
     100x =  27.27272727...

and subtracting the two equations we have

      99x = 27
        x = 27/99 = 3/11

Converting the second fraction by the same process, we have

      10x =  41.666666...
     100x = 416.666666...

and subtracting the two equations we have

      90x = 375
        x = 375/90 = 75/18 = 25/6

Now for an infinite geometric series we have, when r < 1,

     a + ar + ar^2 + ar^3 + .... = a / (1-r)

Evaluating the repeating decimal 0.27272727... using geometric 
series, we have

     0.272727... = 0.27 + 0.0027 + 0.000027 + 0.00000027 + ...
                 = 0.27 + 0.27(.01) + 0.27(.01)^2 + 0.27(.01)^3 + ...
                 = 0.27 / (1-.01)
                 = 0.27 / 0.99
                 = 27/99
                 = 3/11

And evaluating the repeating decimal 4.1666666... using geometric 
series, we have

     4.166666... = 4.1 + .06 + .006 + .0006 + ...
                 = 4.1 + .06 + .06(.1) + .06(.1)^2 + ...
                 = 4.1 + .06 / (1-.1)
                 = 4.1 + .06 / .9
                 = 4.1 + 6/90
                 = 4.1 + 1/15
                 = 41/10 + 1/15
                 = 123/30 + 2/30
                 = 125/30
                 = 25/6

Having written the response above, I just went back and looked at 
your original question, and it seems I may have answered a bigger 
question than the one you asked....

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/