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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.R.63
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)

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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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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.
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Direct 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.
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Exponential Growth & Decay
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 ∞)k⁴ / √(9k¹² + 2)

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²

Textbook Question

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


b.Determine the limit of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

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Textbook Question

Geometric sums

Evaluate the geometric sums

∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

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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 ∞)k⁵ e⁻ᵏ

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 / k³⁄⁷