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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 10.1.67a
Chapter 10, Problem 10.1.67a

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


a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Verified step by step guidance
1
Identify the general term of the series: \( a_k = \frac{2}{(2k - 1)(2k + 1)} \).
Use partial fraction decomposition to rewrite \( a_k \) in a form that allows telescoping. Set \( \frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1} \) and solve for constants \( A \) and \( B \).
Express each partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the decomposed form of \( a_k \) and write out the sum explicitly to observe cancellation of terms.
Calculate the first four partial sums \( S_1, S_2, S_3, S_4 \) by summing the first 1, 2, 3, and 4 terms respectively, using the telescoping form to simplify the sums.
Write each partial sum \( S_n \) in its simplified form after cancellation, which will help in understanding the behavior of the series as \( n \) increases.

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Key Concepts

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

Partial Sums of a Series

Partial sums are the sums of the first n terms of a series, denoted as Sₙ = a₁ + a₂ + ... + aₙ. They help analyze the behavior of infinite series by approximating the total sum and are essential for understanding convergence and series evaluation.
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Intro to Series: Partial Sums

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded, simplifying the sum significantly. Recognizing telescoping patterns allows easier computation of partial sums and limits, often by expressing terms as differences of fractions.
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Geometric Series

Decomposition into Partial Fractions

Partial fraction decomposition breaks a complex rational expression into simpler fractions that are easier to sum or integrate. In series, this technique often reveals telescoping behavior by rewriting terms to highlight cancellations.
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Partial Fraction Decomposition: Distinct Linear Factors
Related Practice
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


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

71. Evaluating an infinite series two ways

Evaluate the series

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

a. Use a telescoping series argument.

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Textbook Question

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


a. Use Sₙ to estimate the sum of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Textbook Question

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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.


a.Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

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Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 20,r = 0.5