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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 10.1.67c
Chapter 10, Problem 10.1.67c

67–70. Formulas for sequences of partial sums Consider the following infinite series.


c.Make a conjecture for the value of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

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Start by examining the general term of the series: \(\frac{2}{(2k - 1)(2k + 1)}\). The goal is to simplify this term to identify a pattern or telescoping behavior.
Use partial fraction decomposition to rewrite the term. Set up the equation: \(\frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1}\), and solve for constants \(A\) and \(B\).
After finding \(A\) and \(B\), express the general term as the difference of two simpler fractions, which will help in telescoping the series when summed from \(k=1\) to \(n\).
Write the partial sum \(S_n = \sum_{k=1}^n \frac{2}{(2k - 1)(2k + 1)}\) using the decomposed form, and observe how most terms cancel out when expanded.
Identify the remaining terms after cancellation to find a formula for \(S_n\), then take the limit as \(n \to \infty\) to conjecture the value of the infinite series.

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Key Concepts

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. To find its value, we must determine if the series converges, meaning its partial sums approach a finite limit. Understanding convergence is essential before assigning a sum to an infinite series.
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Convergence of an Infinite Series

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique helps simplify terms in the series, making it easier to identify patterns or telescoping behavior that can lead to a closed-form sum.
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Partial Fraction Decomposition: Distinct Linear Factors

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded. Recognizing telescoping allows us to simplify the sum of the series by focusing on the first and last terms of the partial sums, facilitating the evaluation of the infinite sum.
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Geometric Series
Related Practice
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


c.Find a recurrence relation that generates the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. A series that converges conditionally must converge.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.


{1, 2, 4, 8, 16, ......}

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Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


c. If ∑ aₖ converges, then ∑ (aₖ + 0.0001) converges.

Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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