Textbook Question
Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.4.50
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (tanh n) / n²
Verified step by step guidance1
Identify the general term of the series: \(a_n = \frac{\tanh n}{n^2}\).
Recall that \(\tanh n\) is the hyperbolic tangent function, which approaches 1 as \(n\) becomes very large, so for large \(n\), \(\tanh n \approx 1\).
Compare the given series to the known convergent p-series \(\sum \frac{1}{n^2}\), which converges because \(p=2 > 1\).
Use the Limit Comparison Test by evaluating \(\lim_{n \to \infty} \frac{a_n}{1/n^2} = \lim_{n \to \infty} \tanh n = 1\), which is a finite nonzero number.
Since the limit is finite and nonzero and \(\sum \frac{1}{n^2}\) converges, conclude that the original series \(\sum \frac{\tanh n}{n^2}\) also converges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like ∑ (tanh n) / n².
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Convergence of an Infinite Series
Comparison Test for Series
The Comparison Test helps determine convergence by comparing a given series to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. This test is useful when dealing with series involving functions like tanh n and polynomial denominators.
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09:25Direct Comparison Test
Behavior of Hyperbolic Tangent Function (tanh n)
The hyperbolic tangent function, tanh n, approaches 1 as n becomes large. Knowing this limit helps simplify the series terms for large n, allowing us to approximate the series and apply convergence tests effectively, especially when combined with terms like 1/n².
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05:13Slopes of Tangent Lines
Related Practice
Textbook Question
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4
Textbook Question
If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.
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Textbook Question
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ
Textbook Question
Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)
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Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 8^(1/n)
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