Textbook Question
18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.
a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.
∑ (k = 0 to ∞) (–2/7)ᵏ
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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 guidance1
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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06:00Geometric 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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10:07Partial Fraction Decomposition: Distinct Linear Factors
Related Practice
Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
b.If a sequence of positive numbers converges, then the sequence is decreasing.
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Textbook Question
41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.
a. Find an upper bound for the remainder in terms of n.
43. ∑ (k = 1 to ∞) 1 / 3ᵏ
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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.
b. Find an explicit formula for the nth term of the sequence {hₙ}.
h₀ = 20,r = 0.5
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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.
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
Find the first term a and the ratio r of each geometric series.
a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ
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