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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.49
Chapter 10, Problem 10.3.49

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…

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1
Identify the repeating decimal part. Here, the repeating block is "037", which repeats indefinitely after the decimal point.
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: \(0.037 + 0.000037 + 0.000000037 + \cdots\).
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 \(a + ar + ar^2 + \cdots\).
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\) to find the sum of the repeating part.
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).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)

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For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


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