9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 2 to ∞) k / ln k

Verified step by step guidance

1. Identify the general term of the series: \(a_k = \frac{k}{\ln k}\) for \(k \geq 2\).
2. Recall the Divergence Test (also known as the Test for Divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
3. Compute the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{\ln k}\).
4. Analyze the behavior of the limit: since \(k\) grows faster than \(\ln k\), the fraction \(\frac{k}{\ln k}\) grows without bound, so the limit does not approach zero.
5. Conclude that by the Divergence Test, since the limit of \(a_k\) is not zero, the series \(\sum_{k=2}^\infty \frac{k}{\ln k}\) diverges.

Key Concepts

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. It is a quick way to check divergence but cannot confirm convergence if the limit is zero.

Behavior of the General Term

Analyzing the behavior of the general term k / ln(k) as k approaches infinity helps determine if the terms approach zero. Since ln(k) grows slower than k, the term k / ln(k) tends to infinity, indicating the terms do not approach zero.

Infinite Series and Convergence

An infinite series converges only if the sum of its terms approaches a finite limit. If the terms do not approach zero, the series cannot converge. Understanding this principle is essential to apply tests like the Divergence Test correctly.