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
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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.R.59
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)!
Verified step by step guidance1
Identify the given series: \( \sum_{j=0}^{\infty} \frac{2 \cdot 4^{j}}{(2j + 1)!} \). We want to determine if this infinite series converges or diverges.
Consider using the Ratio Test, which is often effective for series involving factorials and exponential terms. The Ratio Test states that for \( a_j \), if \( L = \lim_{j \to \infty} \left| \frac{a_{j+1}}{a_j} \right| \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \).
Write the general term \( a_j = \frac{2 \cdot 4^{j}}{(2j + 1)!} \) and the next term \( a_{j+1} = \frac{2 \cdot 4^{j+1}}{(2(j+1) + 1)!} = \frac{2 \cdot 4^{j+1}}{(2j + 3)!} \).
Form the ratio \( \left| \frac{a_{j+1}}{a_j} \right| = \frac{2 \cdot 4^{j+1}}{(2j + 3)!} \times \frac{(2j + 1)!}{2 \cdot 4^{j}} = \frac{4 \cdot (2j + 1)!}{(2j + 3)!} \). Simplify the factorial expression to express the ratio in terms of \( j \).
Evaluate the limit \( L = \lim_{j \to \infty} \left| \frac{a_{j+1}}{a_j} \right| \) using the simplified expression. Based on the value of \( L \), conclude whether the series converges or diverges according to the Ratio Test.
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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 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 the one given.
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Convergence of an Infinite Series
Ratio Test
The Ratio Test is a common method to determine the convergence of series with factorials or exponential terms. It involves taking the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely.
Recommended video:
05:35Ratio Test
Factorials and Growth Rates
Factorials (n!) grow faster than exponential functions as n increases. Recognizing how factorials dominate growth helps in applying convergence tests effectively, especially when terms include factorials in the denominator, often leading to convergence.
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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)
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Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.
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Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)
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Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.
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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?
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