Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 
${\sum }_{n=1}^{\infty }\frac{n^2}{n(n^2-1000)}$

• A. Divergent
• B. Convergent
• C. Inconclusive

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the divergence test, which states that if the limit of the sequence of terms \( a_n \) (the general term of the series) as \( n \to \infty \) does not equal zero, then the series diverges. If the limit equals zero, the test is inconclusive.
2. Step 2: Identify the general term of the series \( a_n = \frac{n^2}{n(n^2 - 1000)} \). Simplify the expression to \( a_n = \frac{n}{n^2 - 1000} \).
3. Step 3: Compute the limit \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n^2 - 1000} \). To evaluate this limit, divide both the numerator and denominator by \( n^2 \), resulting in \( \lim_{n \to \infty} \frac{1/n}{1 - 1000/n^2} \).
4. Step 4: Analyze the behavior of the terms as \( n \to \infty \). The term \( 1/n \to 0 \) and \( 1000/n^2 \to 0 \), so the denominator approaches \( 1 \). Therefore, \( \lim_{n \to \infty} a_n = 0 \).
5. Step 5: Conclude that since \( \lim_{n \to \infty} a_n = 0 \), the divergence test is inconclusive. This means the test does not provide information about whether the series converges or diverges.