# Decimal->Geometric Series

Hi. I'm attempting to convert a decimal to a geometric series of the form ar^n. Does anybody know of any neat little ways to go about doing this? I haven't been able to find anything all that useful on the web yet.

i.e.
4.1666666(6)
.45114141414(14)

Thanks.
Asked:
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Commented:
Ah well.. I do get an idea of what you might be asking now :)

4.16666

= 4.1 + 0.06 + 0.006 + 0.0006 + ...
= 4.1 + Geometric series(a = 0.06, r = 0.1)

For the second,

0.4511414

= .451 + 0.00014 + 0.0000014 + ...
= .451 + Geometric series(a = 0.00014, r = 0.01)
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Commented:
Hi, I don't really understand your question. Can you give an example ?

Let x = 4.16666...
10x = 41.66666...

Subtracting 1 from 2,
9x = 37.5

So, x = 75/18

Hence, any recurring fraction can be put in the form of a fraction.

What do you mean by form of ar^n ?

For your second fraction,

Let x =      .4511414...
100 x = 45.11414...

So, 99x = 44.663

x = 44663/99000

---
Harish
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Commented:
I don't know about geometric series.

Ususally what you want to do with repeating decimals is convert them to fractions:

10*X = 41.66666
X  =  4.16666
---------------------------
9*X = 37.50000       ==>   X  =  37.5/9  =  375/90  =  75/18  =  25/6

For the second one you would have to do

100*Y = 45.1141414
Y  =  0.4511414
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Author Commented:
Hi, I don't really understand your question. Can you give an example ?:

Say for example I wanted to express .080808(08) as the ratio of two integers:

.080808... = 8/(10^2) + 8/(10^4) + 8/(10^6) + 8/(10^8) + ...

sooo we could write that as: Summation(n = 0 -> infinity) (8/(10^2)) (1/(10^2))^n

(Sorry, I don't know how to do summation notation correctly in text. (see above))

So here we have the form ar^n with a = (8/(10^2)) and r = (1/(10^2)). I guess you can use this to find the sum which would be (a/1-r). Anyway, hopefully that example clarified it a bit.

So I guess I'm just trying to apply this same idea to these other two examples whose repeating parts don't start immediately to the right of the decimal as this example does.

Thanks!
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Author Commented:
ahhhh ok I think I've been beating my head against a wall for no reason:

I had something almost identical to this: .451 + Geometric series(a = 0.00014, r = 0.01)

But I thought that the .451 part had to be somehow in the series (which was not making sense to me)...

so I guess my series for the second example would be:

.451 + Summation(n=0,infinity)(14/10^5)(1/.01)^n

?

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Author Commented:
Ok my previous post was wrong:

The series is:

.451 + Summ(n=0,infinity) (14/10^5) (1/100)^n

Thanks Harish! :)
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