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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.53
Capitolo 10, Problema 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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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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Domanda del libro di testo

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

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55–70. More sequences

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


{cosn / n}

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84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.


aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

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What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

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49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.

aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …


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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

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