32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

Verified step by step guidance

1. First, identify the general term of the series: \(a_k = (-1)^k \cdot k \cdot \frac{2^{k+1}}{9^k - 1}\).
2. To determine absolute convergence, consider the absolute value of the terms: \(|a_k| = k \cdot \frac{2^{k+1}}{9^k - 1}\).
3. Analyze the behavior of \(|a_k|\) as \(k\) approaches infinity. Since \$9^k\( grows faster than \)2^{k+1}\(, simplify the expression by comparing dominant terms: \)|a_k| \approx k \cdot \frac{2^{k+1}}{9^k}$.
4. Use the Ratio Test on \(|a_k|\) to check for absolute convergence. Compute the limit \(L = \lim_{k \to \infty} \frac{|a_{k+1}|}{|a_k|}\) and analyze whether \(L < 1\), \(L = 1\), or \(L > 1\).
5. If the series does not converge absolutely, apply the Alternating Series Test to the original series by checking if the sequence \(b_k = k \cdot \frac{2^{k+1}}{9^k - 1}\) is decreasing and if \(\lim_{k \to \infty} b_k = 0\) to determine conditional convergence.

Key Concepts

Absolute and Conditional Convergence

Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the original series converges but the series of absolute values diverges. Distinguishing these helps classify the behavior of alternating or sign-changing series.

Alternating Series Test

The Alternating Series Test determines convergence for series whose terms alternate in sign. It requires that the absolute value of terms decreases monotonically to zero. If these conditions hold, the series converges, but not necessarily absolutely, indicating possible conditional convergence.

Ratio Test

The Ratio Test evaluates the limit of the ratio of consecutive terms' absolute values. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive. This test is especially useful for series involving exponential or factorial terms.