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))
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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.25a
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.
a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.
25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80
Verified step by step guidance1
Identify the given recurrence relation and initial term: \(a_{n+1} = \frac{1}{2} a_n + 8\) with \(a_0 = 80\).
Calculate the first term: \(a_0\) is given as 80.
Find the second term by substituting \(n=0\) into the recurrence: \(a_1 = \frac{1}{2} a_0 + 8\).
Find the third term by substituting \(n=1\): \(a_2 = \frac{1}{2} a_1 + 8\).
Continue this process to find \(a_3 = \frac{1}{2} a_2 + 8\) and \(a_4 = \frac{1}{2} a_3 + 8\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Recurrence Relations
A recurrence relation defines each term of a sequence based on one or more previous terms. In this problem, the sequence is defined recursively by aₙ₊₁ = (1/2) aₙ + 8, meaning each term depends on the previous term. Understanding how to apply the relation iteratively is essential to find specific terms.
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Monotonic and Bounded Sequences
A sequence is monotonic if it is either entirely non-increasing or non-decreasing, and bounded if its terms stay within fixed limits. These properties ensure the sequence behaves predictably and often converges, which helps in analyzing long-term behavior and verifying the correctness of computed terms.
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Iterative Computation of Sequence Terms
To find specific terms like a₀ through a₄, start with the initial term and repeatedly apply the recurrence relation. This step-by-step calculation is fundamental for understanding the sequence's progression and verifying patterns or limits suggested by the recurrence.
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Related Practice
Textbook Question
a.Does the sequence { k/(k + 1) } converge? Why or why not?
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Textbook Question
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
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Textbook Question
What comparison series would you use with the Comparison Test to determine whether
∑ (k = 1 to ∞) 1 / (k² + 1) converges?
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Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²
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Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
27.1 + 1.01 + 1.01² + 1.01³ + ⋯
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