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Advanced Math/Geometric Series & Repeating Decimals



How is a repeating decimal like 0.272727... changed to the fraction 3/11 by using a geometric series?

Please to not omit any steps in your explanation/calculation!

Here is an example take from the following URL:, but I do not understand how the third line becomes = 0.27 / (1-.01).

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

I thank you for your explanation!

The keywords include the formula.
It is from the fact in mathematics on infinite sums.
As stated, it is known that sum(n=0 to ∞)(x^n) = 1/(1-x) if |x|<1.

To display what this mean, let x=0.5.

The equation says for n=0 to n=infinity, stepping by 1, we have
0.5^0 + 0.5^1 + 0.5^2 + 0.5^3 + 0.5^4 + 0.5^5 ...

This is the same as 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + ...

Note that for the first two terms, 1 + 0.5 = 1.5, and that is 2 - 0.5.
Adding the next term 0.25, gives 1.5 + 0.25 = 1.75 = 2 - 0.25 = 2 - 0.5^2.
Adding the next term, 0.125, gives 1.75 + 0.125 = 1.875 = 2 - 0.125 = 2 -0.5^3.
With the next, we get 1.875 + 0.0625 = 1.9375 = 2 - 0.0625 = 2 - 0.5^4.
Adding the next gives 1.9375 + 0.03125 = 1.96875 = 2 - 0.03125 = 2 - 0.5^5.
It can be seen if we go on forever, sum(n=0 to ∞) = 2 - 0.5^∞ = 2 - 0 = 2.

Here, we have x = 0.01, so the sum of 0.01 to the n for n = 0 to infinity  is 1/(1-0.01).

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Scott A Wilson


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