46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
53.0.00952̅ = 0.00952952…

Verified step by step guidance

1. Identify the repeating decimal part. Here, the decimal is 0.00952\overline{952}, meaning the digits '952' repeat indefinitely starting after '0.0095'.
2. Express the decimal as the sum of the non-repeating part and the repeating part. The non-repeating part is 0.0095, and the repeating part starts at the thousandths place with '952' repeating.
3. Write the repeating part as a geometric series. The first term \( a \) corresponds to 0.000952, and the common ratio \( r \) is \( 10^{-3} \) because the block '952' repeats every 3 decimal places. So the series is \( a + ar + ar^2 + \cdots \).
4. Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of the repeating part.
5. Add the non-repeating part to the sum of the repeating part, then simplify the resulting expression to write the entire decimal as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers.

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. Understanding how to identify the repeating part is essential for converting the decimal into a fraction or a series. For example, in 0.00952̅, the digits '952' repeat indefinitely.

Geometric Series Representation

A repeating decimal can be expressed as an infinite geometric series where each term represents the repeating block shifted by powers of 1/10. This series has a first term and a common ratio less than 1, allowing the sum to be calculated using the geometric series formula.

Conversion of Geometric Series to Fraction

The sum of an infinite geometric series with first term a and common ratio r (|r|<1) is a/(1-r). Applying this to the series representing the repeating decimal allows conversion from decimal form to a fraction, expressing the decimal as a ratio of two integers.