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 ∞) 2ᵏ k! / kᵏ

Verified step by step guidance

1. First, write down the general term of the series: \(a_k = \frac{2^k \cdot k!}{k^k}\).
2. To determine absolute convergence, consider the absolute value of the terms, which is the same here since all terms are positive: \(|a_k| = \frac{2^k \cdot k!}{k^k}\).
3. Apply the Root Test or Ratio Test to analyze the behavior of \(a_k\) as \(k\) approaches infinity. The Ratio Test is often convenient for factorials, so compute the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
4. Express \(\frac{a_{k+1}}{a_k}\) explicitly: \(\frac{2^{k+1} (k+1)! / (k+1)^{k+1}}{2^k k! / k^k} = 2 \cdot (k+1) \cdot \frac{k^k}{(k+1)^{k+1}}\).
5. Simplify the expression and evaluate the limit \(L\). If \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive and you may need to try another test.

Key Concepts

Absolute and Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions helps classify the behavior of infinite series.

Ratio Test

The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. It is especially useful for series involving factorials and exponentials.

Factorials and Exponentials in Series

Factorials (k!) grow faster than exponentials (like 2^k), but when combined with terms like k^k, the growth rates affect convergence. Understanding how these terms compare helps in applying tests like the Ratio Test effectively.