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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)]

Verified step by step guidance
1
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


d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

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Textbook Question

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.


Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \(0.


c.How many months are needed to reach a balance of \)5000?

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

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Textbook Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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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. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

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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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