11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 3 to ∞) 5 / (2 + lnk)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=3}^{\infty} \frac{5}{2 + \ln k} \). We want to determine if this series converges or diverges.
2. Consider the behavior of the terms \( a_k = \frac{5}{2 + \ln k} \) as \( k \to \infty \). Since \( \ln k \) grows without bound, analyze the limit of \( a_k \) as \( k \to \infty \).
3. Recall that for a series \( \sum a_k \) to converge, a necessary condition is that \( \lim_{k \to \infty} a_k = 0 \). Check if this condition holds for \( a_k \).
4. If the terms do not approach zero, the series diverges by the Test for Divergence (also called the nth-term test). If they do approach zero, consider applying a comparison test or integral test to determine convergence.
5. Compare \( a_k \) to a simpler series such as \( \sum \frac{1}{\ln k} \) or \( \sum \frac{1}{k} \) to decide convergence. Use the integral test by evaluating \( \int_{3}^{\infty} \frac{5}{2 + \ln x} \, dx \) or compare with a known divergent series to conclude.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Comparison and Limit Comparison Tests

These tests compare a given series to a known benchmark series to determine convergence. The Comparison Test uses inequalities, while the Limit Comparison Test uses limits of term ratios, helping to analyze series with complex terms like those involving logarithms.

Behavior of Logarithmic Functions in Series

Logarithmic functions grow slowly and affect the denominator in series terms. Recognizing how ln(k) behaves as k approaches infinity helps in estimating term sizes and deciding which convergence test is appropriate for series involving logarithms.