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 10.4.43a
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
Verified step by step guidance1
Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{3^k} \) is a geometric series with the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \).
Recall that for a convergent geometric series with \( |r| < 1 \), the sum to infinity is \( S = \frac{a}{1 - r} \).
The remainder \( R_n \) after summing the first \( n \) terms is the difference between the total sum and the partial sum: \( R_n = S - S_n \).
The partial sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1 - r^n}{1 - r} \). Substitute \( a = \frac{1}{3} \) and \( r = \frac{1}{3} \) to express \( S_n \).
Express the remainder \( R_n \) in terms of \( n \) by using the formula \( R_n = S - S_n = \frac{a}{1 - r} - a \frac{1 - r^n}{1 - r} = a \frac{r^n}{1 - r} \). Substitute the values of \( a \) and \( r \) to get the upper bound for the remainder.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergent Geometric Series
A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. If the absolute value of the common ratio is less than 1, the series converges to a finite sum. For example, the series ∑ (1/3)^k converges because |1/3| < 1.
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Geometric Series
Remainder (Error) in Infinite Series
The remainder after n terms of a convergent series is the difference between the infinite sum and the partial sum up to n terms. It measures the error when approximating the infinite sum by a finite sum. Finding an upper bound for the remainder helps estimate how close the partial sum is to the total.
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Formula for Remainder in Geometric Series
For a geometric series with first term a and common ratio r (|r|<1), the remainder after n terms is given by R_n = a * r^(n+1) / (1 - r). This formula provides an explicit upper bound for the error when approximating the infinite sum by the first n terms.
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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)]
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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
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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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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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