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)

검증된 단계별 안내

Identify the series given: \( \sum_{k=1}^{\infty} \frac{k^{2}}{4^{k}} \). We want to determine if it converges absolutely or diverges.

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.

Set \( a_k = \frac{k^{2}}{4^{k}} \). Compute \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{2} / 4^{k+1}}{k^{2} / 4^{k}} = \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} \).

Take the limit as \( k \to \infty \) of \( \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} = \left( \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \right) \cdot \frac{1}{4} \).

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.

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주요 개념

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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.

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)