Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) k⁸ / (k¹¹ + 3)
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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 10.3.37
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37.1 + e/π + e²/π² + e³/π³ + ⋯
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Identify the first term \( a \) of the geometric series. Here, the first term is \( 1 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{e/\pi}{1} = \frac{e}{\pi} \).
Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| \frac{e}{\pi} \right| \). The series converges if \( |r| < 1 \) and diverges otherwise.
If the series converges, use the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio.
Substitute the values of \( a \) and \( r \) into the formula to express the sum of the series.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form a + ar + ar² + ar³ + ⋯, where a is the first term and r is the common ratio.
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Geometric Series
Convergence of Geometric Series
A geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges and does not have a finite sum.
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Evaluating the Given Series
To evaluate the series 1 + e/π + e²/π² + ⋯, identify the first term a = 1 and the common ratio r = e/π. Since e ≈ 2.718 and π ≈ 3.1415, |r| < 1, so the series converges. Use the sum formula S = 1 / (1 - e/π) to find the sum.
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Related Practice
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯
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Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
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Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (2k + 1)! / (k!)²
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Textbook Question
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…
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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
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