40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) (5lnk) / k

검증된 단계별 안내

Identify the series given: \( \sum_{k=2}^{\infty} \frac{5 \ln k}{k} \). We want to determine if this series converges or diverges.

Since the terms involve \( \ln k \) and \( k \), consider using the Integral Test, which is suitable for series with positive, continuous, and decreasing terms for large \( k \).

Set up the corresponding integral for the Integral Test: \( \int_{2}^{\infty} \frac{5 \ln x}{x} \, dx \).

Evaluate the integral by using substitution: let \( u = \ln x \), then \( du = \frac{1}{x} dx \), so the integral becomes \( 5 \int u \, du \).

Determine whether the integral converges or diverges by evaluating the limit as the upper bound approaches infinity. If the integral converges, the series converges; if it diverges, the series diverges.

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주요 개념

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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. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the partial sums stabilize.

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series. The Comparison Test checks if terms are smaller or larger than a convergent or divergent series, while the Limit Comparison Test uses the limit of the ratio of terms to determine convergence behavior.

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, provided f is positive, continuous, and decreasing.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.∑ (k = 2 to ∞) (5lnk) / k