Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.
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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.63
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)3 / (2 + eᵏ)
Verified step by step guidance1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{3}{2 + e^{k}} \). We want to determine if this infinite series converges or diverges.
Observe the general term of the series: \( a_k = \frac{3}{2 + e^{k}} \). Since \( e^{k} \) grows exponentially, the denominator increases very quickly as \( k \) becomes large.
Compare \( a_k \) to a simpler series to test for convergence. Notice that for large \( k \), \( 2 + e^{k} \approx e^{k} \), so \( a_k \approx \frac{3}{e^{k}} \). This suggests comparing to the geometric series \( \sum \frac{3}{e^{k}} \).
Recall that a geometric series \( \sum r^{k} \) converges if \( |r| < 1 \). Here, \( r = \frac{1}{e} \), which is less than 1, so \( \sum \frac{3}{e^{k}} \) converges.
By the Comparison Test, since \( 0 < \frac{3}{2 + e^{k}} < \frac{3}{e^{k}} \) for all \( k \) and \( \sum \frac{3}{e^{k}} \) converges, the original series \( \sum_{k=1}^{\infty} \frac{3}{2 + e^{k}} \) also converges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite limit) or diverges (grows without bound or oscillates) is fundamental in calculus. Convergence ensures the series has a meaningful sum.
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06:52
Convergence of an Infinite Series
Comparison Test for Series
The Comparison Test determines convergence by comparing the given series to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
Recommended video:
09:25Direct Comparison Test
Exponential Growth and Its Impact on Series Terms
Exponential functions like e^k grow very rapidly as k increases. In the series terms 3/(2 + e^k), the denominator grows exponentially, causing terms to approach zero quickly, which is a key factor in assessing convergence.
Recommended video:
09:29Exponential Growth & Decay
Related Practice
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
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!
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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
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Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)
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Textbook Question
Building a tunnel — first scenario
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.
b.What is the longest tunnel the crew can build at this rate?
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