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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 10.5.50
Chapter 10, Problem 10.5.50

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) (5lnk) / k

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1
Identify the series given: \( \sum_{k=2}^{\infty} \frac{5 \ln k}{k} \). We want to determine if this series converges or diverges.
Since the terms involve \( \ln k \) and \( k \), consider using the Integral Test, which is suitable for series with positive, continuous, and decreasing terms for large \( k \).
Set up the corresponding integral for the Integral Test: \( \int_{2}^{\infty} \frac{5 \ln x}{x} \, dx \).
Evaluate the integral by using substitution: let \( u = \ln x \), then \( du = \frac{1}{x} dx \), so the integral becomes \( 5 \int u \, du \).
Determine whether the integral converges or diverges by evaluating the limit as the upper bound approaches infinity. If the integral converges, the series converges; if it diverges, the series diverges.

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Key Concepts

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

Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the partial sums stabilize.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series. The Comparison Test checks if terms are smaller or larger than a convergent or divergent series, while the Limit Comparison Test uses the limit of the ratio of terms to determine convergence behavior.
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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 f(x) from some point to infinity converges, then the series ∑ f(k) also converges, provided f is positive, continuous, and decreasing.
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Integral Test
Related Practice
Textbook Question

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

∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰⁰⁰ / 2ⁿ}

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Textbook Question

Is it possible for an alternating series to converge absolutely but not conditionally?

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Textbook Question

Periodic doses

Suppose you take 200 mg of an antibiotic every 6 hr. The half-life of the drug (the time it takes for half of the drug to be eliminated from your blood) is 6 hr. Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood. You may assume the steady-state amount is finite.

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Textbook Question

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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Textbook Question

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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