9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((-1)ᵏ) / (k!)

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \). Since the terms involve factorials, the Ratio Test is often convenient here.
2. Recall the Ratio Test formula: For the series \( \sum a_k \), compute \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.
3. Write the general term \( a_k = \frac{(-1)^k}{k!} \). Then, \( a_{k+1} = \frac{(-1)^{k+1}}{(k+1)!} \). Compute the ratio of absolute values: \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{(-1)^{k+1}}{(k+1)!} \cdot \frac{k!}{(-1)^k} \right| = \frac{1}{k+1} \).
4. Evaluate the limit as \( k \to \infty \): \( L = \lim_{k \to \infty} \frac{1}{k+1} = 0 \). Since \( L < 1 \), the series converges absolutely.
5. Conclude that by the Ratio Test, the series \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k!} \) converges absolutely because the limit of the ratio of consecutive terms is zero.

Key Concepts

Ratio Test

The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Root Test

The Root Test analyzes the nth root of the absolute value of the terms in a series. If the limit of this nth root as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence, and it is a stronger form of convergence that allows the use of tests like the Ratio and Root Tests effectively.