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
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 10.3.71a
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.
Verified step by step guidance1
Write out the general term of the series explicitly: \(a_k = \frac{4}{3^k} - \frac{4}{3^{k+1}}\).
Recognize that this is a telescoping series because each term can be rewritten to show cancellation between consecutive terms when summed.
Express the partial sum \(S_n = \sum_{k=1}^n \left( \frac{4}{3^k} - \frac{4}{3^{k+1}} \right)\) and write out the first few terms to observe the telescoping pattern.
Notice that most terms cancel out, leaving only the first term of the first fraction and the last term of the second fraction in the partial sum.
Write the simplified form of \(S_n\) after cancellation and then take the limit as \(n \to \infty\) to find the sum 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.
Telescoping Series
A telescoping series is a series where many terms cancel out when the partial sums are expanded, leaving only a few terms to evaluate. This simplification makes it easier to find the sum of the infinite series by focusing on the first and last terms of the partial sums.
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06:00
Geometric Series
Infinite Geometric Series
An infinite geometric series has terms that multiply by a constant ratio each time. If the absolute value of the ratio is less than one, the series converges, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
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06:00Geometric Series
Partial Sums and Convergence
Partial sums are the sums of the first n terms of a series. Understanding how these sums behave as n approaches infinity helps determine if the series converges and what its sum is. For telescoping and geometric series, analyzing partial sums is key to evaluating the infinite sum.
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08:01Integration Using Partial Fractions
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
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)]
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
{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