11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k \frac{k}{k^3 + 1} \). Notice that it is an alternating series because of the factor \( (-1)^k \).
2. Consider the absolute value of the terms: \( a_k = \frac{k}{k^3 + 1} \). Simplify or analyze the behavior of \( a_k \) as \( k \to \infty \).
3. Check the limit of \( a_k \) as \( k \to \infty \): \( \lim_{k \to \infty} \frac{k}{k^3 + 1} \). This will help determine if the terms approach zero, which is necessary for convergence of an alternating series.
4. Apply the Alternating Series Test (Leibniz Test), which requires that \( a_k \) is positive, decreasing, and \( \lim_{k \to \infty} a_k = 0 \). Verify these conditions for \( a_k \).
5. If the Alternating Series Test confirms convergence, consider whether the series converges absolutely by testing the convergence of \( \sum_{k=1}^{\infty} \left| (-1)^k \frac{k}{k^3 + 1} \right| = \sum_{k=1}^{\infty} \frac{k}{k^3 + 1} \) using a comparison or limit comparison test with a known convergent or divergent series.

Key Concepts

Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. If the absolute value of terms decreases monotonically to zero, the series converges. This test is useful for series like ∑ (−1)^k * a_k where a_k > 0 and lim a_k = 0.

Limit of the General Term

For any infinite series ∑ a_k to converge, the terms a_k must approach zero as k approaches infinity. If lim (a_k) ≠ 0, the series diverges. Checking this limit is a fundamental first step in analyzing series convergence.

Comparison Test

The Comparison Test compares a given series to a known benchmark series to determine convergence. If the terms of the given series are smaller than those of a convergent series, it converges; if larger than a divergent series, it diverges. This helps analyze the behavior of complex terms.