9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)

검증된 단계별 안내

Identify the given series: \( \sum_{k=4}^{\infty} \frac{1 + \cos^{2}(k)}{k - 3} \). We want to determine if this series converges or diverges.

Analyze the behavior of the terms for large \(k\). Notice that \(1 + \cos^{2}(k)\) oscillates between 1 and 2 because \(\cos^{2}(k)\) is always between 0 and 1.

Choose a comparison series to apply the Comparison Test or Limit Comparison Test. Since the denominator is \(k - 3\), which behaves like \(k\) for large \(k\), consider the series \( \sum_{k=4}^{\infty} \frac{1}{k} \), which is a harmonic series known to diverge.

Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{\frac{1 + \cos^{2}(k)}{k - 3}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{(1 + \cos^{2}(k)) \cdot k}{k - 3} \). Simplify this expression to understand the behavior of \(L\).

Since \(1 + \cos^{2}(k)\) oscillates but stays bounded between 1 and 2, the limit \(L\) will be a finite positive number. By the Limit Comparison Test, the original series behaves like the harmonic series \(\sum \frac{1}{k}\), which diverges. Therefore, conclude about the convergence or divergence of the original series.

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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 a second series with known behavior. If the terms of the given series are less than or equal to the terms of a convergent series, it also converges. Conversely, if the terms are greater than or equal to 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 for large indices.

Behavior of Trigonometric Functions in Series

Understanding the bounded nature of trigonometric functions like cosine is crucial. Since cos²(k) oscillates between 0 and 1, the numerator (1 + cos²(k)) stays between 1 and 2, allowing simplification in comparison tests. Recognizing this helps in estimating the series terms and choosing an appropriate comparison series.

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)