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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 10.3.53
Chapter 10, Problem 10.3.53

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…

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1
Identify the repeating decimal part. Here, the decimal is 0.00952\(\overline{952}\), meaning the digits '952' repeat indefinitely starting after '0.0095'.
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.
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 \( a + ar + ar^2 + \cdots \).
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of the repeating part.
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.

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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. 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.
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Repeated Integration by Parts

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.
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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.
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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 / (2k + 1)

Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))

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Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰⁰⁰ / 2ⁿ}

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Textbook Question

Is it possible for an alternating series to converge absolutely but not conditionally?

Textbook Question

Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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Textbook Question

54–69. Telescoping series

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.


63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer

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