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)
76
views
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.95b
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?
Verified step by step guidance1
Recognize that the distance dug each week forms a geometric sequence where the first term \(a_1 = 100\) meters and the common ratio \(r = 0.95\) because each week the crew digs 0.95 times the previous week's distance.
Recall that the total length of the tunnel after infinitely many weeks is the sum of an infinite geometric series, which converges when \(|r| < 1\).
Use the formula for the sum of an infinite geometric series: \(S_\infty = \frac{a_1}{1 - r}\), where \(a_1\) is the first term and \(r\) is the common ratio.
Substitute the known values into the formula: \(S_\infty = \frac{100}{1 - 0.95}\) to express the total maximum length of the tunnel.
Interpret this sum as the longest tunnel the crew can build at this decreasing rate of construction.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is the sum of terms where each term is a constant multiple (common ratio) of the previous one. In this problem, the weekly distances dug form a geometric sequence with ratio 0.95. Understanding how to sum an infinite geometric series helps find the total tunnel length when the digging rate decreases continuously.
Recommended video:
06:00
Geometric Series
Sum to Infinity of a Geometric Series
When the common ratio of a geometric series is between -1 and 1, the series converges to a finite sum. The formula S = a / (1 - r) calculates this sum, where 'a' is the first term and 'r' is the ratio. This concept is essential to determine the maximum tunnel length achievable as the digging rate diminishes over time.
Recommended video:
06:00Geometric Series
Modeling Real-World Problems with Sequences
Translating a real-world scenario into a mathematical sequence allows for precise analysis. Here, the weekly digging distances decrease by a fixed percentage, forming a geometric sequence. Recognizing this pattern enables the use of sequence formulas to predict total progress and solve practical construction problems.
Recommended video:
06:16Real World Application
Related Practice
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.
39
views
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)
65
views
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 ∞)3 / (2 + eᵏ)
65
views
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.
40
views
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)!
35
views