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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.6.76
Chapter 10, Problem 10.6.76

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(ln n / n)³]

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Identify the given series: \( \sum_{n=2}^{\infty} \left( \frac{\ln n}{n} \right)^3 \). We want to determine if this series converges or diverges.
Recognize that the terms are positive for \( n \geq 2 \), so we can consider convergence tests for series with positive terms.
Consider the Comparison Test or Limit Comparison Test by comparing \( \left( \frac{\ln n}{n} \right)^3 \) to a simpler series. Since \( \ln n \) grows slower than any power of \( n \), compare it to \( \frac{1}{n^p} \) for some \( p \).
Rewrite the term as \( \frac{(\ln n)^3}{n^3} \). Since \( (\ln n)^3 \) grows slower than any power of \( n \), the dominant factor is \( \frac{1}{n^3} \).
Use the p-series test: since \( \sum \frac{1}{n^3} \) converges (because \( p=3 > 1 \)), and \( \left( \frac{\ln n}{n} \right)^3 \) behaves like \( \frac{(\ln n)^3}{n^3} \), the original series converges by the Comparison Test.

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Key Concepts

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

Convergence and Divergence of Series

A series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Understanding this helps determine whether the infinite sum has a meaningful value or not.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare a given series to a known benchmark series to determine convergence or divergence. If terms of the series behave like those of a convergent or divergent series, the original series shares the same behavior.
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07:45
Limit Comparison Test

Behavior of Logarithmic and Polynomial Terms in Series

Analyzing how terms like (ln n)³ and n³ grow helps in assessing the series' terms. Since logarithmic functions grow slower than polynomial functions, this relationship is crucial in applying comparison tests effectively.
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Taylor Series
Related Practice
Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

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Textbook Question

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

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Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

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Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) sin (1/n)

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Textbook Question

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 5,aₙ₊₁ = √(5aₙ)

Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ