Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁵ e⁻ᵏ
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14. Sequences & Series
Convergence Tests
2:34 minutesProblem 10.5.19
Textbook Question
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)
Verified step by step guidance1
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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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Direct Comparison Test
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.
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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.
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