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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 10.3.59
Chapter 10, Problem 10.3.59

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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1
Start by expressing the general term of the series: \( a_k = \frac{4}{(4k - 3)(4k + 1)} \). Our goal is to rewrite this term in a form that reveals telescoping behavior.
Use partial fraction decomposition to break \( a_k \) into simpler fractions. Assume \( \frac{4}{(4k - 3)(4k + 1)} = \frac{A}{4k - 3} + \frac{B}{4k + 1} \) and solve for constants \( A \) and \( B \).
Once you find \( A \) and \( B \), rewrite \( a_k \) as the difference of two fractions, which will help terms cancel out when summing from \( k = -3 \) to \( n \).
Write the partial sum \( S_n = \sum_{k=-3}^n a_k \) using the decomposed form. Observe how most terms cancel out due to the telescoping nature, leaving only a few terms from the start and end of the sum.
Express \( S_n \) explicitly in terms of \( n \), then analyze \( \lim_{n \to \infty} S_n \) to determine whether the series converges or diverges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Telescoping Series

A telescoping series is a series whose partial sums simplify by canceling intermediate terms, leaving only a few terms from the beginning and end. This simplification makes it easier to find a formula for the nth partial sum and analyze convergence.
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Partial Sums and Their Formulas

The partial sum Sₙ of a series is the sum of its first n terms. Finding a closed-form expression for Sₙ helps in evaluating the behavior of the series as n approaches infinity, which is essential for determining convergence or divergence.
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Limit of a Sequence

The limit of the sequence of partial sums, limₙ→∞ Sₙ, determines the sum of an infinite series if it exists. If this limit is finite, the series converges; otherwise, it diverges. Understanding limits is crucial for evaluating infinite series.
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Related Practice
Textbook Question

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

∑ (k = 1 to ∞) cos(k) / k³

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Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k

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Textbook Question

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


∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ

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Textbook Question

55–70. More sequences

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


{cosn / n}

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Textbook Question

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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Textbook Question

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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