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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 10.4.19
Chapter 10, Problem 10.4.19

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))

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First, identify the function corresponding to the terms of the series: \( f(x) = \frac{1}{\sqrt[3]{5x + 3}} \). This function is defined for \( x \geq 1 \).
Check the conditions for the Integral Test: verify that \( f(x) \) is positive, continuous, and decreasing for \( x \geq 1 \). Since the cube root function is continuous and increasing, and the denominator \( 5x + 3 \) is increasing, \( f(x) \) is positive and continuous. To check if \( f(x) \) is decreasing, consider the derivative \( f'(x) \) and verify it is negative for \( x \geq 1 \).
Set up the improper integral corresponding to the series: \( \int_1^{\infty} \frac{1}{\sqrt[3]{5x + 3}} \, dx \).
Evaluate the integral by using an appropriate substitution, such as \( u = 5x + 3 \), which simplifies the integral to a form involving \( u^{-1/3} \).
Determine whether the improper integral converges or diverges by evaluating the limit as the upper bound approaches infinity. If the integral converges, then by the Integral Test, the series \( \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{5k + 3}} \) also converges; if it diverges, the series diverges as well.

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Key Concepts

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

Integral Test for Convergence

The Integral Test determines the convergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing, then the series converges if and only if the integral of the function from a certain point to infinity converges.
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Choosing a Convergence Test

Conditions for Applying the Integral Test

To apply the Integral Test, the function f(x) representing the series terms must be positive, continuous, and decreasing for all x greater than or equal to some number N. Verifying these conditions ensures the test's validity and that the behavior of the integral reflects the series' behavior.
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Integral Test

Evaluating Improper Integrals

Evaluating the improper integral involves integrating the function from a finite point to infinity and determining if the limit exists and is finite. This process often requires substitution and limit evaluation techniques to conclude whether the integral converges or diverges.
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Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (k² / 4ᵏ)

Textbook Question

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^(–1/5)

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 ∞) (2k + 1)! / (k!)²

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


61. ∑ (k = 1 to ∞) ln((k + 1) / k)

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Textbook Question

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

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 / k⁵

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