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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 10.5.45
Chapter 10, Problem 10.5.45

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 3 to ∞) 1 / lnk

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Identify the series given: \( \sum_{k=3}^{\infty} \frac{1}{\ln k} \). We want to determine if this series converges or diverges.
Since the terms involve a logarithmic function in the denominator, consider using the Integral Test, which is suitable for positive, continuous, and decreasing functions for \( k \geq 3 \).
Define the function \( f(x) = \frac{1}{\ln x} \) for \( x \geq 3 \). Verify that \( f(x) \) is positive and decreasing on this interval.
Set up the improper integral \( \int_{3}^{\infty} \frac{1}{\ln x} \, dx \) to analyze the behavior of the series using the Integral Test.
Evaluate or analyze the integral \( \int_{3}^{\infty} \frac{1}{\ln x} \, dx \). If the integral diverges, then the series \( \sum_{k=3}^{\infty} \frac{1}{\ln k} \) also diverges; if it converges, then the series converges.

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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 increases 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 Test and Limit Comparison Test

These tests compare the given series to a known benchmark series. If the terms of the given series behave similarly to a convergent or divergent series, the original series shares the same convergence behavior. The limit comparison test uses the limit of the ratio of terms to establish this relationship.
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Limit Comparison Test

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, and vice versa. This test is useful when terms involve functions like logarithms.
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Related Practice
Textbook Question

What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


37.1 + e/π + e²/π² + e³/π³ + ⋯

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Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

47.0.3̅ = 0.333…

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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/k)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from j = 1 to ∞) 5 / (j² + 4)

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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