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 ∞) (k² / 4ᵏ)

Verified step by step guidance

1. Identify the series given: \( \sum_{k=1}^{\infty} \frac{k^{2}}{4^{k}} \). We want to determine if it converges absolutely or diverges.
2. Recall the Ratio Test: For a series \( \sum a_k \), compute the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.
3. Set \( a_k = \frac{k^{2}}{4^{k}} \). Compute \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{2} / 4^{k+1}}{k^{2} / 4^{k}} = \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} \).
4. Take the limit as \( k \to \infty \) of \( \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} = \left( \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \right) \cdot \frac{1}{4} \).
5. Evaluate the limit \( \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} = 1 \), so the overall limit \( L = 1 \cdot \frac{1}{4} = \frac{1}{4} \). Since \( L < 1 \), by the Ratio Test, the series converges absolutely.

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 nth term of a series. If the limit of this root is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for series with terms raised to the nth power.

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence, ensuring stability under rearrangement of terms.