9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k² + 4)

검증된 단계별 안내

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

Choose a comparison series that is simpler but behaves similarly for large \( k \). Since \( k^2 + 4 \) behaves like \( k^2 \) for large \( k \), consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 \).

Recall that the p-series \( \sum \frac{1}{k^p} \) converges if \( p > 1 \). Since \( p = 2 > 1 \), the comparison series \( \sum \frac{1}{k^2} \) converges.

Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{\frac{1}{k^2 + 4}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{k^2}{k^2 + 4} \).

Evaluate the limit \( L \). If \( L \) is a finite positive number, then both series either both converge or both diverge. Since the comparison series converges, this will imply the original series converges as well.

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주요 개념

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Comparison Test

The Comparison Test determines the convergence of a series by comparing it to another series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar term behavior.

p-Series and Their Convergence

A p-series has the form ∑ 1/k^p and converges if p > 1, diverging otherwise. Recognizing that 1/(k² + 4) behaves like 1/k² for large k helps apply comparison tests effectively, since the p-series with p=2 is known to converge.

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.∑ (k = 1 to ∞) 1 / (k² + 4)