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

Verified step by step guidance

1. 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.
2. 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 \( k \geq 2 \).
3. 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.
4. 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 \( \int_{\ln 2}^{\infty} \frac{1}{u} \, du \).
5. 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.

Key Concepts

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.