9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

Verified step by step guidance

1. Identify the general term of the series: \(a_k = \frac{k}{2k + 1}\).
2. Recall the Divergence Test (also known as the nth-term test for divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
3. Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{2k + 1}\).
4. To find this limit, divide numerator and denominator by \(k\): \(\lim_{k \to \infty} \frac{1}{2 + \frac{1}{k}}\).
5. Evaluate the limit: as \(k \to \infty\), \(\frac{1}{k} \to 0\), so the limit becomes \(\frac{1}{2}\). Since this limit is not zero, by the Divergence Test, the series 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. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index k becomes very large. Evaluating this limit helps determine the behavior of the series' terms, which is essential for applying the Divergence Test.

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges (does not approach a finite value) is fundamental in calculus, often requiring tests like the Divergence Test.