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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.R.75
Chapter 10, Problem 10.R.75

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)tanh(k)

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1
Identify the series given: \( \sum_{k=1}^{\infty} \tanh(k) \). We want to determine if this infinite series converges or diverges.
Recall that for a series \( \sum a_k \) to converge, the terms \( a_k \) must approach zero as \( k \to \infty \). So, first examine the behavior of \( \tanh(k) \) as \( k \to \infty \).
Note that \( \tanh(k) = \frac{e^{k} - e^{-k}}{e^{k} + e^{-k}} \). As \( k \to \infty \), \( e^{k} \) dominates \( e^{-k} \), so \( \tanh(k) \to 1 \).
Since the terms \( \tanh(k) \) do not approach zero but instead approach 1, the necessary condition for series convergence is not met.
Therefore, by the Test for Divergence (also called the nth-term test), the series \( \sum_{k=1}^{\infty} \tanh(k) \) diverges.

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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(k).
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Convergence of an Infinite Series

Behavior of the Hyperbolic Tangent Function (tanh)

The hyperbolic tangent function, tanh(k), approaches 1 as k becomes very large. Since its terms do not approach zero, this behavior is critical in assessing the convergence of the series ∑ tanh(k), because terms must approach zero for the series to have a chance to converge.
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Slopes of Tangent Lines

Divergence Test (Nth-Term Test)

The divergence test states that if the limit of the terms of a series does not approach zero, the series diverges. This is a quick and effective test to determine divergence, especially useful here since lim(k→∞) tanh(k) = 1 ≠ 0, implying the series diverges.
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Divergence Test (nth Term Test)
Related Practice
Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 1 to ∞)ln((2k + 1) / (2k − 1))

Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)(1 − cos(1 / k))²

Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 0 to ∞)((1/3)ᵏ + (4/3)ᵏ)

Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k! / (eᵏ kᵏ)

Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from j = 0 to ∞)2 ⋅ 4ʲ / (2j + 1)!

Textbook Question

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.


b.What is the longest tunnel the crew can build at this rate?