45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) cos(k) / k³

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{\cos(k)}{k^3} \). We want to determine if it converges absolutely, conditionally, or diverges.

Recall that absolute convergence means the series \( \sum |a_k| \) converges, where \( a_k = \frac{\cos(k)}{k^3} \). So consider the series \( \sum_{k=1}^{\infty} \left| \frac{\cos(k)}{k^3} \right| = \sum_{k=1}^{\infty} \frac{|\cos(k)|}{k^3} \).

Since \( |\cos(k)| \leq 1 \) for all integers \( k \), the terms \( \frac{|\cos(k)|}{k^3} \) are bounded above by \( \frac{1}{k^3} \). We know that \( \sum_{k=1}^{\infty} \frac{1}{k^3} \) is a p-series with \( p = 3 > 1 \), which converges.

By the Comparison Test, since \( 0 \leq \frac{|\cos(k)|}{k^3} \leq \frac{1}{k^3} \) and \( \sum \frac{1}{k^3} \) converges, the series \( \sum \frac{|\cos(k)|}{k^3} \) converges. Therefore, the original series converges absolutely.

Since absolute convergence implies convergence, there is no need to check for conditional convergence. The series \( \sum_{k=1}^{\infty} \frac{\cos(k)}{k^3} \) converges absolutely.

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Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the terms' signs, and it often simplifies analysis by allowing the use of comparison tests on positive terms.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating or oscillating series where sign changes affect convergence.

Comparison and Limit Comparison Tests

These tests help determine convergence by comparing a given series to a known benchmark series. For example, comparing ∑|cos(k)/k³| to ∑1/k³, a p-series with p=3, which converges, can establish absolute convergence of the original series.

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.∑ (k = 1 to ∞) cos(k) / k³