Textbook Question
Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.
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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.R.3
Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.
Verified step by step guidance1
Recognize that both sums are geometric series where each term is of the form \(r^k\) with common ratio \(r = 0.2\).
Recall the formula for the sum of the first \(n+1\) terms of a geometric series starting at \(k=0\): \(S = \frac{1 - r^{n+1}}{1 - r}\).
For the first sum \(\sum_{k=0}^{9} (0.2)^k\), identify \(n=9\) and apply the formula: \(S_1 = \frac{1 - (0.2)^{10}}{1 - 0.2}\).
For the second sum \(\sum_{k=2}^{9} (0.2)^k\), express it as the difference between the sum from \(k=0\) to \(9\) and the sum from \(k=0\) to \(1\): \(S_2 = \sum_{k=0}^{9} (0.2)^k - \sum_{k=0}^{1} (0.2)^k\).
Calculate \(\sum_{k=0}^{1} (0.2)^k\) using the geometric sum formula with \(n=1\): \(S_{0\text{ to }1} = \frac{1 - (0.2)^2}{1 - 0.2}\), then subtract this from \(S_1\) to find \(S_2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is the sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure is essential for evaluating sums like those given.
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Geometric Series
Formula for the Sum of a Finite Geometric Series
The sum of the first n+1 terms of a geometric series is given by S = a(1 - r^(n+1)) / (1 - r), where a is the first term and r is the common ratio (r ≠ 1). This formula allows quick calculation of sums without adding each term individually.
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Index Shifting in Summations
When the summation index does not start at zero, it is often helpful to rewrite the sum by shifting the index to start at zero. This simplifies applying the geometric series formula by adjusting the first term and the number of terms accordingly.
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Related Practice
Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹
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Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a.The terms of the sequence {aₙ} increase in magnitude, so the limit of the sequence does not exist.
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Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ
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Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)2ᵏ / eᵏ
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Textbook Question
Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.
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