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))
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.R.71
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
Verified step by step guidance1
First, write down the series explicitly: \( \sum_{k=1}^{\infty} \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \). Our goal is to determine whether this infinite series converges or diverges.
Recall that for large \(k\), \( \frac{1}{k} \) is close to zero. We can use the Taylor series expansion of \( \cos x \) around \( x = 0 \) to approximate \( \cos\left(\frac{1}{k}\right) \). The expansion is \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots \).
Substitute \( x = \frac{1}{k} \) into the expansion to approximate \( 1 - \cos\left(\frac{1}{k}\right) \). This gives \( 1 - \cos\left(\frac{1}{k}\right) \approx \frac{1}{2k^2} - \frac{1}{24k^4} + \cdots \).
Since the series term is \( \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \), square the approximation to find the leading term behavior. The dominant term will be proportional to \( \frac{1}{k^4} \) because \( \left(\frac{1}{2k^2}\right)^2 = \frac{1}{4k^4} \).
Compare the series \( \sum \left(1 - \cos\left(\frac{1}{k}\right)\right)^2 \) to the p-series \( \sum \frac{1}{k^4} \). Since \( p = 4 > 1 \), the p-series converges, so by the Comparison Test, the original series converges as well.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity or oscillate indefinitely.
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Convergence of an Infinite Series
Comparison and Limit Comparison Tests
These tests compare the given series to a known benchmark series to determine convergence. The Comparison Test uses inequalities, while the Limit Comparison Test uses the limit of the ratio of terms, helping to infer convergence or divergence by relating to simpler, well-understood series.
Recommended video:
07:45Limit Comparison Test
Asymptotic Behavior and Series Term Approximation
Analyzing the behavior of terms for large indices often involves approximations like Taylor expansions. For example, approximating cos(1/k) for large k helps simplify terms to known forms, enabling the use of standard convergence tests on the approximated series.
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06:45Intro to Series: Partial Sums
Related Practice
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 ∞)tanh(k)
Textbook Question
89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.
b. Find an upper bound for the remainder Rₙ.
89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5
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
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?