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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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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

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = sin x,a = 0

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Textbook Question

Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

Textbook Question

Use series to evaluate the limits in Exercises 29–40.

29. lim (x → 0) (e^x - (1 + x)) / x²

Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)

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Textbook Question

Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

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Textbook Question

In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.

1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

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