Textbook Question
Evaluate 1000!/998! without a calculator.
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 10.7.29
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 ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)
Verified step by step guidance1
Identify the general term of the series as \(a_k = \frac{(-1)^{k+1} \cdot k^{2k}}{(k!) \cdot (k!)}\).
Since the series has alternating signs, to check for absolute convergence, consider the absolute value of the terms: \(|a_k| = \frac{k^{2k}}{(k!)^2}\).
Apply the Ratio Test by computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1)^{2(k+1)}}{((k+1)!)^2} \cdot \frac{(k!)^2}{k^{2k}}\).
Simplify the expression inside the limit by expressing factorials and powers explicitly, for example, \((k+1)! = (k+1) \cdot k!\), and rewrite powers to compare terms.
Evaluate the limit \(L\) and use the Ratio Test criteria: if \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ratio Test
The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Ratio Test
Root Test
The Root Test analyzes the nth root of the absolute value of the terms in a series. If the limit of this nth root as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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07:15Root Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence regardless of the sign of terms, which is important when dealing with alternating series like the given one.
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07:51Choosing a Convergence Test
Related Practice
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45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞)k² · 1.001⁻ᵏ
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72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)
Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (2k − √k)