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### 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

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/
```
Associated Topics:
High School Sequences, Series
Middle School Fractions

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