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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.R.39
Chapter 10, Problem 10.R.39

Express 0.314141414… as a ratio of two integers.

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1
Identify the repeating decimal pattern in the number 0.314141414… Here, the digits '14' repeat indefinitely after the initial '3'.
Let \( x = 0.314141414\ldots \). To isolate the repeating part, multiply \( x \) by a power of 10 that moves the decimal point just before the repeating block. Since '14' has 2 digits, multiply by 100: \( 100x = 31.4141414\ldots \).
Next, multiply \( x \) by a power of 10 that moves the decimal point just before the first repeating block starts after the non-repeating part. Since the non-repeating part is '3' (one digit), multiply by 10: \( 10x = 3.14141414\ldots \).
Subtract the two equations to eliminate the repeating decimal: \( 100x - 10x = 31.4141414\ldots - 3.14141414\ldots \). This simplifies to \( 90x = 28.273\ldots \), but since the decimals after subtraction cancel out, you get \( 90x = 28.273\ldots \) with the repeating part removed.
Solve for \( x \) by dividing both sides by 90, then simplify the resulting fraction to express the original decimal as a 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 Decimal Representation

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. Recognizing the repeating part is essential to convert the decimal into a fraction. For example, in 0.314141414…, the digits '14' repeat indefinitely after the initial '3'.
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Repeated Integration by Parts

Algebraic Method for Converting Repeating Decimals to Fractions

This method involves setting the repeating decimal equal to a variable, multiplying by powers of 10 to shift the decimal point, and subtracting to eliminate the repeating part. Solving the resulting equation yields the fraction form. This approach systematically converts infinite decimals into ratios of integers.
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Partial Fraction Decomposition: Repeated Linear Factors Example 4

Simplification of Fractions

After finding the fraction equivalent of a repeating decimal, simplifying it by dividing numerator and denominator by their greatest common divisor ensures the fraction is in lowest terms. This step is important for expressing the ratio in its simplest and most understandable form.
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Related Practice
Textbook Question

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯


89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.


a. Find the base-10 representation of the binary number 0.001̅1.

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Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

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Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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Textbook Question

Give an example (if possible) of a sequence {aₖ} that converges, while the series ∑ (from k = 1 to ∞) aₖ diverges.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. n!n! = (2n)! for all positive integers n.

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