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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.1.53
Chapter 10, Problem 10.1.53

51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.


aₙ₊₁ = 4aₙ + 1 a₀ = 1

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Identify the recurrence relation given: \(a_{n+1} = 4a_n + 1\) with initial term \(a_0 = 1\).
Create a table of terms by calculating each subsequent term using the recurrence relation: for each \(n\), compute \(a_{n+1}\) by substituting \(a_n\) into the formula \(a_{n+1} = 4a_n + 1\).
Calculate the first ten terms step-by-step: start with \(a_0 = 1\), then find \(a_1 = 4 \times a_0 + 1\), \(a_2 = 4 \times a_1 + 1\), and so on until \(a_9\).
Observe the behavior of the terms in the table: check if the terms are increasing without bound, approaching a fixed number, or oscillating.
Based on the observed pattern, determine whether the sequence converges to a limit or diverges (grows without bound).

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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 way to generate sequences step-by-step, often starting from an initial value. Understanding how to apply and iterate these relations is essential for analyzing the behavior of sequences.
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Sequence Limits and Convergence

The limit of a sequence is the value the terms approach as the index goes to infinity. A sequence converges if its terms get arbitrarily close to a finite number; otherwise, it diverges. Determining limits helps understand the long-term behavior of sequences defined by recurrence relations.
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Iterative Computation and Pattern Recognition

Computing terms iteratively involves using the recurrence formula repeatedly to build a sequence table. Observing these computed values helps identify patterns or trends, which is crucial for hypothesizing about the sequence’s limit or divergence before formal proof.
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Related Practice
Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(−1)ⁿ / 2ⁿ}

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Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

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

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)

Textbook Question

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.