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In this question it must convert the repeating decimal 0.576923076923076923…. into fraction with the help of geometric series.

Repeating Decimal to Fraction:

A fraction is a number that represents a portion of a whole in mathematics. It is, in other words, a two-number ratio. In contrast, a decimal is a number with a decimal point between the total number and the fractional sections. There are two kinds of decimal numbers: terminating and non-terminating decimals, as well as repeating and non-repeating decimals. When dealing with many mathematical problems, converting decimal to fractional value is favoured because fractional values may be readily simplified.

The repeating decimal can be written as:

$\begin{array}{rcl}0.576923076923076923...& =& 0.5+0.076923+0.000000076923+.......\\ & =& \frac{1}{2}+\frac{76923}{{10}^6}+\frac{76923}{{10}^{12}}+...\\ & =& \frac{1}{2}+\underset{i=1}{\overset{∞}{∑}}\frac{76923}{{10}^{6i}}\\ & =& \frac{1}{2}+\underset{i=1}{\overset{∞}{∑}}\frac{76923}{{1000000}^i}\\ & & \end{array}$

Further solve the above expression,

$\begin{array}{rcl}0.576923076923076923...& =& \frac{1}{2}+76923\underset{i=1}{\overset{∞}{∑}}\frac{1}{{1000000}^i}\\ & =& \frac{1}{2}+76923\underset{i=1}{\overset{∞}{∑}}{\left(\frac{1}{1000000}\right)}^i\\ & =& \frac{1}{2}+76923\left[-1+\underset{i=1}{\overset{∞}{∑}}{\left(\frac{1}{1000000}\right)}^i\right]\\ & & \text{=}\frac{1}{2}+76923\left[-1+\frac{1}{1-\left(\frac{1}{1000000}\right)}\right]\\ & & \end{array}$

Therefore, solve the above expression

$$\begin{gathered} 0.576923076923076923...\text{=}\frac{1}{2}+76923\left(-1+\frac{1000000}{999999}\right) \\ \text{=}\frac{1}{2}+76923\left(\frac{1}{999999}\right) \\ =\frac{15}{26} \end{gathered}$$

Hence, the required answer is $0.576923076923076923...=\frac{15}{26}$.