Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.4.33

Chapter 10, Problem 10.4.33

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ

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Identify the series given: \( \sum_{k=1}^{\infty} \frac{k}{e^{k}} \). Notice that the terms involve \( k \) in the numerator and an exponential \( e^{k} \) in the denominator.

Consider the behavior of the terms \( a_k = \frac{k}{e^{k}} \) as \( k \to \infty \). Since the denominator grows exponentially and the numerator grows linearly, the terms \( a_k \) approach zero, which is a necessary condition for convergence.

Apply the Divergence Test first: check if \( \lim_{k \to \infty} a_k \neq 0 \). If the limit is not zero, the series diverges. Here, the limit is zero, so the Divergence Test is inconclusive.

Use the Integral Test or compare with a known convergent series. Since \( e^{k} \) grows faster than any polynomial, compare \( \frac{k}{e^{k}} \) to \( \frac{1}{e^{k/2}} \) or recognize it as a series with terms decreasing exponentially.

Conclude that the series converges by comparison to a convergent geometric series or by applying the Integral Test to the function \( f(x) = \frac{x}{e^{x}} \), which is positive, continuous, and decreasing for \( x \geq 1 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.

Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of the continuous, positive, decreasing function from which the series terms are derived converges, then the series converges as well.

p-series Test

The p-series Test applies to series of the form ∑ 1/n^p. Such a series converges if and only if p > 1, and diverges otherwise. It is useful for comparing or identifying the behavior of series with terms involving powers of n.

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Related Practice

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) sin(1 / k) / k²

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Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ / k⁰.⁹⁹

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Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{√((1 + 1 / 2n)ⁿ)}

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Textbook Question

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.

∑ (k = 1 to ∞) (−1)ᵏ / kᵏ

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Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / (2√k − 1)

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Textbook Question

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.∑ (k = 1 to ∞) k / eᵏ

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Key Concepts

Divergence Test

Integral Test

p-series Test

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k / eᵏ