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 10.3.5a
Find the first term a and the ratio r of each geometric series.
a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ
Verified step by step guidance1
Identify the general form of a geometric series, which is given by the sum from k = 0 to infinity of \( a \times r^k \), where \( a \) is the first term and \( r \) is the common ratio.
Compare the given series \( \sum_{k=0}^\infty \left( \frac{2}{3} \times \left( \frac{1}{5} \right)^k \right) \) to the general form to determine the first term \( a \) and the ratio \( r \).
Note that the first term \( a \) corresponds to the term when \( k = 0 \), which is \( \frac{2}{3} \times \left( \frac{1}{5} \right)^0 = \frac{2}{3} \times 1 = \frac{2}{3} \).
Identify the common ratio \( r \) as the factor raised to the power \( k \), which is \( \frac{1}{5} \).
Summarize that the first term \( a = \frac{2}{3} \) and the common ratio \( r = \frac{1}{5} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series Definition
A geometric series is a sum of terms where each term after the first is found by multiplying the previous term by a constant ratio r. It can be written as a + ar + ar² + ar³ + ... , where a is the first term and r is the common ratio.
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Geometric Series
Identifying the First Term (a)
The first term a of a geometric series is the initial term when the index k is zero. In the series ∑ from k=0 to ∞ of a × r^k, the first term is simply the coefficient multiplied by r raised to the zero power, which equals a.
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Common Ratio (r)
The common ratio r is the factor by which each term is multiplied to get the next term. It is the base of the exponent k in the series expression a × r^k, and it determines the behavior and convergence of the series.
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Related Practice
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
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
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
{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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