Textbook Question
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 3ᵏ⁺² / 5ᵏ
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 10.1.23
"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 3aₙ-12; a₁ = 10
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Identify the given recurrence relation: \(a_{n+1} = 3a_n - 12\) with the initial term \(a_1 = 10\).
Calculate the second term \(a_2\) by substituting \(n=1\) into the recurrence relation: \(a_2 = 3a_1 - 12\).
Calculate the third term \(a_3\) by substituting \(n=2\): \(a_3 = 3a_2 - 12\).
Calculate the fourth term \(a_4\) by substituting \(n=3\): \(a_4 = 3a_3 - 12\).
List the first four terms of the sequence as \(a_1\), \(a_2\), \(a_3\), and \(a_4\) after computing each step.
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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 using previous terms. It provides a formula to generate terms step-by-step, starting from initial values. Understanding how to apply the relation repeatedly is essential to find subsequent terms.
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Initial Conditions
Initial conditions specify the starting values of a sequence, such as a₁ = 10. These values are necessary to begin the process of generating terms using the recurrence relation, as they anchor the sequence and allow computation of all following terms.
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05:03Initial Value Problems
Iterative Computation of Terms
To find terms of a sequence defined by a recurrence relation, you substitute known terms into the relation repeatedly. This iterative process involves calculating each term step-by-step, using the previous term(s) and the given formula.
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Related Practice
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−7)ᵏ / k!
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³
Textbook Question
8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.
∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4
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Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = (−1)ⁿ ⁿ√n
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}