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
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.29
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)
Verified step by step guidance1
Identify the given series as the sum of two separate infinite series: \( \sum_{k=0}^\infty \left( \left(\frac{1}{3}\right)^k + \left(\frac{4}{3}\right)^k \right) = \sum_{k=0}^\infty \left(\frac{1}{3}\right)^k + \sum_{k=0}^\infty \left(\frac{4}{3}\right)^k \).
Recognize that each series is a geometric series of the form \( \sum_{k=0}^\infty r^k \), where \( r \) is the common ratio.
Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). If it converges, the sum is given by \( \frac{1}{1-r} \).
Check the common ratios: for the first series, \( r = \frac{1}{3} \), which satisfies \( |r| < 1 \), so it converges; for the second series, \( r = \frac{4}{3} \), which does not satisfy \( |r| < 1 \), so it diverges.
Conclude that since the second series diverges, the entire original series diverges and does not have a finite sum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Geometric Series
An infinite geometric series is a sum of terms where each term is obtained by multiplying the previous term by a constant ratio. It converges if the absolute value of the ratio is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
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Geometric Series
Convergence and Divergence of Series
A series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the terms do not approach zero or the sum grows without bound, the series diverges. Determining convergence is essential before evaluating the sum.
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Sum of a Series with Multiple Terms
When a series is composed of sums of multiple sequences, the sum of the series is the sum of the individual series, provided each converges. This allows breaking down complex series into simpler parts that can be evaluated separately and then combined.
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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
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
89–90. {Use of Tech} Lower and upper bounds of a series
For each convergent series and given value of n, complete the following.
c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.
89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5
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?