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A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. We often encounter geometric series in mathematics when dealing with repeating decimals.

For example, the repeating decimal portion of the number in our exercise, which is 0.0083 repeated, can be expressed as the infinite geometric series: \[\begin{equation}0.\bar{83} = 83\left(\frac{1}{100}\right)^1 + 83\left(\frac{1}{100}\right)^2 + 83\left(\frac{1}{100}\right)^3 + \cdots\end{equation}\]Each term in this series is a product of the initial term, 83, and a power of \(\frac{1}{100}\), making it a textbook example of a geometric series.

The sum of an infinite geometric series can be determined using a specific formula, provided the series converges, meaning the absolute value of the common ratio is less than 1. The formula is given by \[\begin{equation}S = \frac{a}{1 - r}\end{equation}\]where \(S\) is the sum of the series, \(a\) is the first term, and \(r\) is the common ratio. In the context of our repeating decimal, we applied this formula to find that the sum of the repeating portion \(0.\bar{83}\) is \(\frac{83}{99}\). This elegant technique allows us to transform something seemingly infinite and complex into a simple, rational number.

Expressing a decimal as a fraction involves identifying the repeating and non-repeating parts of the decimal, if any, and then converting each into fraction form. The non-repeating part is straightforward to convert, as we just count the number of decimal places, place the decimal as the numerator, and the corresponding power of 10 as the denominator.

For the repeating part, as in the example \(0.\bar{83}\), we can use the geometric series method to write it as a fraction. This two-pronged approach lets us translate any decimal—repeating or not—into a fraction, which can be a more manageable form, especially for further mathematical operations.

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator are the smallest possible integers that have the same ratio as the original fraction. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by that number.

For instance, in our problem, the final fraction obtained was \(\frac{58956}{9900}\). By finding the GCD of 58956 and 9900, we can simplify the fraction down to \(\frac{14789}{2475}\), its simplest form. Simplifying fractions makes them easier to understand and work with, as it gives us the most reduced version of the ratio, eliminating any factor common to both the numerator and the denominator.