Textbook Question
Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.
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
27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)
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 ∞)5ᵏ / 2²ᵏ⁺¹
Textbook Question
{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?
Textbook Question
Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.
1
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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 = 3 to ∞)ln(k) / k³ᐟ²