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A geometric series is a sum of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Let's explore this concept further by examining what makes it so special in the context of repeating decimals.

The series we're dealing with in this scenario is the repeating part of the decimal, expressed as a geometric series. For instance, the repeating decimal \(0.\overline{54}\) becomes \(0.545454\ldots\), which is often represented as:

• \(\frac{54}{100} + \frac{54}{100^2} + \frac{54}{100^3} + \ldots\)

Here, each term is a smaller power of 100, creating a clear pattern. The first term or initial term \(a\) is \(\frac{54}{100}\) while the common ratio \(r\) is \(\frac{1}{100}\).
When these terms are added indefinitely, they form an infinite geometric series, which is where we turn to our next topic.

An infinite series is a sum of terms that continues indefinitely. When discussing repeating decimals, the infinite series comes in to show how these repeating parts stretch infinitely, forming an endless sequence.

In an infinite geometric series, if the absolute value of the common ratio \(r\) is less than 1, the sum \(S\) of the series can be found using the formula:
\[ S = \frac{a}{1-r} \]
where \(a\) is the first term and \(r\) is the common ratio.

For the repeating decimal \(2.\overline{54}\), only the sequence \(0.545454\ldots\) forms an infinite series. Here:

• First term \(a = \frac{54}{100}\)


• Common ratio \(r = \frac{1}{100}\)

Plug these into the formula to find the sum of the infinite series:
\[ S = \frac{\frac{54}{100}}{1 - \frac{1}{100}} = \frac{\frac{54}{100}}{\frac{99}{100}} = \frac{54}{99} = \frac{6}{11} \].
This fractional result of the infinite series provides the crucial repeating part for converting the full decimal.

Transforming repeating decimals into fractions is a helpful skill, especially with infinite series at your disposal. Once you see the pattern of repetition, you can turn an endless sequence into a neat fraction.

Let's take the example of the decimal \(2.\overline{54}\). We've already learned how to represent its repeating part \(0.\overline{54}\) as fractions by using geometric series. The repeating decimal translated to \(\frac{6}{11}\), which we solved in the infinite series section.
When you combine this with the non-repeating whole number 2, you can express the entire decimal as a fraction:

• The whole number part is \(2\), equivalently \(\frac{22}{11}\).
• Add that to \(\frac{6}{11}\) to get \(\frac{28}{11}\).

This conversion shows that repeating decimals, even though they seem complex, can be systematically broken down into fractions, making them much easier to understand and work with.