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.9a
Sequences versus series
a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.
Verified step by step guidance1
Identify the given sequence as \( a_k = \left(-\frac{4}{5}\right)^k \), where \( k \) is a positive integer.
Recall that the limit of a sequence \( a_k \) as \( k \to \infty \) depends on the behavior of the base \( r = -\frac{4}{5} \) raised to the power \( k \).
Since \( |r| = \left| -\frac{4}{5} \right| = \frac{4}{5} < 1 \), the terms \( r^k \) approach zero as \( k \to \infty \).
Consider the effect of the negative sign: the terms alternate in sign but their magnitude decreases to zero, so the sequence converges to zero.
Therefore, the limit of the sequence \( \left\{ \left(-\frac{4}{5}\right)^k \right\} \) as \( k \to \infty \) is zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Their Limits
A sequence is an ordered list of numbers defined by a specific formula for its terms. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine the long-term behavior of sequences.
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8:22
Introduction to Sequences
Geometric Sequences
A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. The general form is a_k = a * r^k, where r is the common ratio. Recognizing geometric sequences simplifies finding limits and sums.
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04:18Geometric Sequences - Recursive Formula
Limit of a Geometric Sequence with |r| < 1
If the absolute value of the common ratio r in a geometric sequence is less than 1, the terms approach zero as k approaches infinity. This property is crucial for evaluating limits of sequences like (−4/5)^k, where |−4/5| = 0.8 < 1.
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04:18Geometric Sequences - Recursive Formula
Related Practice
Textbook Question
25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.
b.Determine the limit of each sequence.
25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80
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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 ∞)k⁵ e⁻ᵏ
Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k³⁄⁷
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?