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 ∞) 1 / (1000 + k)

Verified step by step guidance

1. Identify the general term of the series, which is given by \(a_k = \frac{1}{1000 + k}\).
2. Recall the Divergence Test (also known as the nth-term test for divergence), which states: If \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges. If the limit equals zero, the test is inconclusive.
3. Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{1}{1000 + k}\).
4. Evaluate the limit by considering the behavior of the denominator as \(k\) becomes very large.
5. Based on the value of the limit, conclude whether the Divergence Test shows the series diverges or if the test is inconclusive.

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 or divergence.

Behavior of the General Term

Analyzing the general term of the series, 1/(1000 + k), involves understanding its limit as k approaches infinity. Since the denominator grows without bound, the term approaches zero, which means the Divergence Test cannot confirm divergence in this case.

Comparison to Harmonic Series

The series ∑ 1/(1000 + k) resembles the harmonic series ∑ 1/k, which is known to diverge. Recognizing this similarity helps in concluding that despite terms approaching zero, the series diverges by comparison to a known divergent series.