40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 2 to ∞) 1 / (klnk)

검증된 단계별 안내

Identify the series given: \( \sum_{k=2}^{\infty} \frac{1}{k \ln k} \). Notice that the terms are positive and involve a logarithmic function in the denominator.

Recognize that this series resembles a p-series or a series that can be tested using the Integral Test because the terms are positive, continuous, and decreasing for \( k \geq 2 \).

Set up the Integral Test by considering the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \). This integral will help determine the convergence of the series.

Evaluate or analyze the integral \( \int_{2}^{\infty} \frac{1}{x \ln x} \, dx \) by using the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} dx \). This transforms the integral into \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \).

Determine the behavior of the integral \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \). Since this integral diverges (it behaves like the harmonic integral), conclude about the convergence or divergence of the original series based on the Integral Test.

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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. Understanding convergence is essential to determine whether the series sums to a finite value or diverges to infinity.

Integral Test

The integral test compares a series to an improper integral to determine convergence. If the integral of the corresponding continuous, positive, decreasing function converges, then the series converges; otherwise, it diverges. This test is useful for series involving functions like 1/(k ln k).

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series. The comparison test uses inequalities, while the limit comparison test uses limits of term ratios. They help determine convergence by relating the series to simpler, well-understood series such as p-series or harmonic series.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.∑ (k = 2 to ∞) 1 / (klnk)