Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
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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 10.7.7
What test is advisable if a series involves a factorial term?
Verified step by step guidance1
When a series involves factorial terms, it is often advisable to use the Ratio Test to determine convergence or divergence.
The Ratio Test involves examining the limit of the absolute value of the ratio of consecutive terms: \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
For a series with terms involving factorials, this test simplifies the factorial expressions because factorials cancel nicely when taking ratios of consecutive terms.
If the limit from the Ratio Test is less than 1, the series converges absolutely; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.
Therefore, the Ratio Test is the most effective and commonly used test for series with factorial terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorial Terms in Series
Factorial terms, denoted by n!, grow very rapidly and often appear in series related to permutations or combinations. Recognizing factorials in series helps determine the appropriate convergence test, as their growth rate affects the behavior of the series.
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Factorials
Ratio Test
The Ratio Test is used to determine the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. It is especially effective for series with factorials or exponential terms because it simplifies the factorial expressions and reveals convergence behavior.
Recommended video:
05:35Ratio Test
Convergence of Infinite Series
Understanding when an infinite series converges or diverges is fundamental in calculus. Convergence means the sum approaches a finite value, and tests like the Ratio Test help decide this, particularly when terms involve complex expressions like factorials.
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Related Practice
Textbook Question
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-7)ᵏ / k²)
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)
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Textbook Question
Evaluate 1000!/998! without a calculator.
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Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
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Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
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