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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.R.83
Capitolo 10, Problema 10.R.83

77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ

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Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k k e^{-k} \). This is an alternating series because of the factor \( (-1)^k \).
To check for absolute convergence, consider the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| (-1)^k k e^{-k} \right| = \sum_{k=1}^{\infty} k e^{-k} \).
Analyze the absolute value series \( \sum_{k=1}^{\infty} k e^{-k} \). Since \( e^{-k} = \frac{1}{e^k} \), the terms look like \( \frac{k}{e^k} \). Use a convergence test suitable for series with terms involving \( k \) and exponential decay, such as the Ratio Test.
Apply the Ratio Test to \( a_k = k e^{-k} \): compute \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1) e^{-(k+1)}}{k e^{-k}} \) and simplify the expression to determine if the limit is less than 1.
If the absolute value series converges, then the original series converges absolutely. If it does not, check if the original alternating series converges by applying the Alternating Series Test, which requires that \( k e^{-k} \) decreases to zero as \( k \to \infty \).

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Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or ratio tests.
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Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.
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Tests for Convergence of Series

To determine convergence, tests like the Ratio Test, Root Test, and Alternating Series Test are used. The Ratio Test is useful for series with exponential terms, while the Alternating Series Test applies to series with alternating signs and decreasing terms. These tests help classify the series as absolutely convergent, conditionally convergent, or divergent.
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Percorso guidato
07:51
Choosing a Convergence Test
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Domanda del libro di testo

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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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

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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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b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

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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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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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