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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 10.5.57
Chapter 10, Problem 10.5.57

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) 1 / (klnk)

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1
Identify the series given: \( \sum_{k=2}^{\infty} \frac{1}{k \ln k} \). Notice that the terms are positive and involve a logarithmic function in the denominator.
Recognize that this series resembles a p-series or a series that can be tested using the Integral Test because the terms are positive, continuous, and decreasing for \( k \geq 2 \).
Set up the Integral Test by considering the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \). This integral will help determine the convergence of the series.
Evaluate or analyze the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \) by using the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} dx \). This transforms the integral into \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \).
Determine the behavior of the integral \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \). Since this integral diverges (it behaves like the harmonic integral), conclude about the convergence or divergence of the original series based on the Integral Test.

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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. Understanding convergence is essential to determine whether the series sums to a finite value or diverges to infinity.
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Convergence of an Infinite Series

Integral Test

The integral test compares a series to an improper integral to determine convergence. If the integral of the corresponding continuous, positive, decreasing function converges, then the series converges; otherwise, it diverges. This test is useful for series involving functions like 1/(k ln k).
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Integral Test

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series. The comparison test uses inequalities, while the limit comparison test uses limits of term ratios. They help determine convergence by relating the series to simpler, well-understood series such as p-series or harmonic series.
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Limit Comparison Test
Related Practice
Textbook Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

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.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

Textbook Question

38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.

39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

Textbook Question

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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Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = 8ⁿ / n!

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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(k² + 4) / (2k² + 1)