Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.67b

Chapter 10, Problem 10.1.67b

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

b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Verified step by step guidance

Start by writing the nth partial sum of the series as the sum of the first n terms: \(S_n = \sum_{k=1}^n \frac{2}{(2k - 1)(2k + 1)}\).

Use partial fraction decomposition to rewrite the general term \(\frac{2}{(2k - 1)(2k + 1)}\) in a form that will allow telescoping. Set \(\frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1}\) and solve for A and B.

After finding A and B, rewrite each term of the sum using these partial fractions. This should create a telescoping series where most terms cancel out when summed.

Express \(S_n\) as a simplified expression after cancellation, which will be a formula involving only a few terms depending on n.

Use the formula for \(S_n\) to calculate the next four partial sums: \(S_5\), \(S_6\), \(S_7\), and \(S_8\) by substituting n = 5, 6, 7, and 8 respectively.

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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

A partial sum Sₙ is the sum of the first n terms of a series. It helps approximate the value of an infinite series by adding a finite number of terms. Understanding partial sums is essential to analyze convergence and to find explicit formulas for sums up to a certain index.

Telescoping Series

A telescoping series is one where many terms cancel out when the partial sums are expanded. This simplification often allows finding a closed-form expression for Sₙ. Recognizing telescoping patterns is key to simplifying complex series like the given one.

Decomposition into Partial Fractions

Partial fraction decomposition breaks a complex rational expression into simpler fractions. This technique is useful to rewrite terms in a series to reveal telescoping behavior. Applying it to the given term helps in finding a formula for the nth partial sum.

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Related Practice

Textbook Question

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

b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.

43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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.

b. Find an explicit formula for the nth term of the sequence {hₙ}.

h₀ = 20,r = 0.5

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Textbook Question

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

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

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

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Textbook Question

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

b. Evaluate the series using Theorem 10.7.

∑ (k = 0 to ∞) (–2/7)ᵏ

Textbook Question

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

b. A series that converges absolutely must converge.

Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

b.Find an explicit formula for the terms of 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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