Textbook Question
What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.3.47
46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
47.0.3̅ = 0.333…
Verified step by step guidance1
Identify the repeating decimal: here it is \(0.3\overline{3} = 0.333\ldots\) where the digit 3 repeats indefinitely.
Express the repeating decimal as an infinite geometric series. The first term is \(\frac{3}{10}\) (since 3 is in the tenths place), and each subsequent term is multiplied by \(\frac{1}{10}\) because the decimal repeats every place value. So the series is: \(\frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \cdots\).
Recognize this as a geometric series with first term \(a = \frac{3}{10}\) and common ratio \(r = \frac{1}{10}\).
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(|r| < 1\).
Substitute \(a = \frac{3}{10}\) and \(r = \frac{1}{10}\) into the formula to write the decimal as a fraction: \(S = \frac{\frac{3}{10}}{1 - \frac{1}{10}}\).
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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 digit or group of digits repeats infinitely. For example, 0.3̅ means 0.333..., where the digit 3 repeats endlessly. Understanding this helps in converting such decimals into exact fractions.
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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 expressed as infinite geometric series, where each term represents a decimal place value.
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Conversion of Infinite Geometric Series to Fractions
An infinite geometric series with first term 'a' and common ratio 'r' (|r|<1) converges to a/(1-r). This formula allows us to convert the geometric series representation of a repeating decimal into a fraction, providing an exact ratio of two integers.
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Related Practice
Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37.1 + e/π + e²/π² + e³/π³ + ⋯
1
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Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk
Textbook Question
23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k^(1/k)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)