9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{4 \ln k} \). Note that the term for \(k=1\) is undefined since \(\ln 1 = 0\), so consider the series starting from \(k=2\) instead.
2. Recall that the Comparison Test and Limit Comparison Test are used to determine convergence by comparing the given series to a known benchmark series. Since the terms involve \( \frac{1}{\ln k} \), consider comparing it to the harmonic series \( \sum \frac{1}{k} \), which is divergent.
3. Set up the Limit Comparison Test by defining \(a_k = \frac{1}{4 \ln k} \) and \(b_k = \frac{1}{k} \). Compute the limit \( L = \lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{1/(4 \ln k)}{1/k} = \lim_{k \to \infty} \frac{k}{4 \ln k} \).
4. Analyze the limit \( L = \lim_{k \to \infty} \frac{k}{4 \ln k} \). Since \(k\) grows faster than \(\ln k\), this limit tends to infinity, which means the Limit Comparison Test with \(b_k = \frac{1}{k}\) is inconclusive.
5. Try a different comparison series, such as \(b_k = \frac{1}{k^p} \) for some \(p > 0\), or consider the integral test for \( \int \frac{1}{\ln x} dx \) to analyze convergence. The integral test can help determine if the series converges or diverges by examining the behavior of the integral of the function \(f(x) = \frac{1}{4 \ln x} \) for \(x \geq 2\).

Key Concepts

Comparison Test

The Comparison Test determines the convergence or divergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than those of a divergent series, it diverges.

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together, making it useful when direct comparison is difficult.

Behavior of the Harmonic Series and Logarithmic Terms

Understanding the divergence of the harmonic series and how logarithmic terms affect convergence is crucial. Since ∑ 1/k diverges, series like ∑ 1/(ln k) require careful comparison, as logarithmic growth is slow and affects the series' convergence properties.