Final answer:
A repeating decimal can be expressed as a geometric series and then converted into a fraction using the sum of an infinite geometric series formula.
Step-by-step explanation:
A repeating decimal can be written as a geometric series by expressing it as the sum of an infinite geometric sequence. To find the fraction representation, we can use the formula for the sum of an infinite geometric series.
For example, let's take the repeating decimal 0.333... To express it as a geometric series, we can write it as 0.3 + 0.03 + 0.003 + ... This is an infinite geometric sequence with a common ratio of 0.1 (each term is 1/10 of the previous term).
Now, to find the fraction representation, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
In this case, a = 0.3 and r = 0.1. Plugging these values into the formula, we get S = 0.3 / (1 - 0.1) = 0.3 / 0.9 = 1/3.
Learn more about Converting repeating decimals to fractions