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
Verified step by step guidance1
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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2mKey Concepts
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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
a.Does the sequence { k/(k + 1) } converge? Why or why not?
1
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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 = (–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