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.
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.51
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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Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^k}{e^k} \). This is a series where each term is \( \frac{2^k}{e^k} \).
Rewrite the general term to recognize the type of series: \( \frac{2^k}{e^k} = \left( \frac{2}{e} \right)^k \). This shows the series is geometric with common ratio \( r = \frac{2}{e} \).
Recall the convergence criterion for a geometric series: A geometric series \( \sum r^k \) converges if and only if \( |r| < 1 \).
Evaluate the absolute value of the common ratio: \( \left| \frac{2}{e} \right| \). Since \( e \approx 2.718 \), compare \( 2 \) and \( e \) to determine if \( |r| < 1 \).
Based on the comparison, conclude whether the series converges or diverges by applying the geometric series test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series
An infinite series is the sum of infinitely many terms, often expressed as ∑ a_k from k=1 to ∞. Understanding whether such a series converges (approaches a finite limit) or diverges (grows without bound or oscillates) is fundamental in calculus.
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Convergence of an Infinite Series
Geometric Series and Ratio Test
A geometric series has terms of the form ar^k. The Ratio Test compares the limit of |a_(k+1)/a_k| to 1; if less than 1, the series converges absolutely. This test is especially useful for series with exponential terms like 2^k/e^k.
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Exponential Functions and Their Growth Rates
Exponential functions like 2^k and e^k grow at different rates. Since e ≈ 2.718, e^k grows faster than 2^k, which affects the behavior of the terms and helps determine convergence by comparing numerator and denominator growth.
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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
Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)
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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
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)(7 + sin k) / k²
Textbook Question
Estimate the value of the series ∑ (from k = 1 to ∞)1 / (2k + 5)³ to within 10⁻⁴ of its exact value.