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 ∞) ((-7)ᵏ / k²)

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k^2} \). We want to determine if it converges absolutely or diverges.
2. Recall that absolute convergence means the series \( \sum_{k=1}^{\infty} \left| a_k \right| \) converges, where \( a_k = \frac{(-7)^k}{k^2} \). So consider the series \( \sum_{k=1}^{\infty} \frac{7^k}{k^2} \).
3. Apply the Ratio Test, which uses the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). For our terms, calculate \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{(-7)^{k+1} / (k+1)^2}{(-7)^k / k^2} \right| = \frac{7^{k+1}}{7^k} \cdot \frac{k^2}{(k+1)^2} = 7 \cdot \left( \frac{k}{k+1} \right)^2 \).
4. Evaluate the limit as \( k \to \infty \): \( L = 7 \cdot \lim_{k \to \infty} \left( \frac{k}{k+1} \right)^2 = 7 \cdot 1 = 7 \).
5. Interpret the result of the Ratio Test: since \( L = 7 > 1 \), the series \( \sum_{k=1}^{\infty} \frac{7^k}{k^2} \) diverges, so the original series does not converge absolutely. Because the terms do not tend to zero in absolute value, the original series diverges.

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 guarantees convergence regardless of the sign of terms, which is important when applying tests like the Ratio or Root Test to series with alternating or negative terms.