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]

검증된 단계별 안내

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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주요 개념

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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.

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.

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.

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]