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


49.0.037̅ = 0.037037…

Verified step by step guidance

1. Identify the repeating decimal part. Here, the repeating block is "037", which repeats indefinitely after the decimal point.
2. Express the decimal as a sum of its non-repeating part plus the repeating part written as an infinite geometric series. Since the repeating block "037" starts immediately after the decimal, write it as: \$0.037 + 0.000037 + 0.000000037 + \cdots$.
3. Rewrite the repeating part as a geometric series with the first term \(a = 0.037\) and common ratio \(r = 10^{-3}\) (because the repeating block has 3 digits, so each term is shifted by 3 decimal places). 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. Combine the sum of the geometric series with any non-repeating part (if any) and simplify the expression to write the decimal as a fraction (ratio of two integers).

Key Concepts

Repeating Decimals

A repeating decimal is a decimal number in which a sequence of digits repeats infinitely. For example, 0.037037… has the block '037' repeating endlessly. Understanding this pattern is essential to express the decimal as a series or fraction.

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Repeating decimals can be represented as infinite geometric series by identifying the repeating block as the first term and the power of 10 as the common ratio.

Conversion of Geometric Series to Fraction

An infinite geometric series with first term a and common ratio r (|r|<1) sums to a/(1-r). Using this formula, the repeating decimal's geometric series can be converted into a fraction, expressing the decimal as a ratio of two integers.