Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)
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14. Sequences & Series
Convergence Tests
2:18 minutesProblem 10.5.25
Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (2k − √k)
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{2k - \sqrt{k}} \). We want to determine if this series converges or diverges.
To apply the Comparison Test or Limit Comparison Test, find a simpler series to compare with. Notice that for large \(k\), \(2k - \sqrt{k} \approx 2k\), so the terms behave like \( \frac{1}{2k} \).
Consider the comparison series \( \sum_{k=1}^{\infty} \frac{1}{k} \), which is a harmonic series known to diverge.
Use the Limit Comparison Test by computing \( \lim_{k \to \infty} \frac{\frac{1}{2k - \sqrt{k}}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{k}{2k - \sqrt{k}} \). Simplify this limit to determine if it is a finite positive number.
If the limit is a positive finite constant, then both series either both converge or both diverge. Since the harmonic series diverges, conclude the behavior of the original series accordingly.
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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 smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than 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 terms behave similarly for large indices.
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Behavior of Series Terms and Dominant Terms
Analyzing the dominant terms in the series' general term helps simplify the expression for large indices. For example, in 1/(2k - √k), the term 2k dominates √k as k grows large, so the series behaves like 1/(2k). Understanding this helps in choosing an appropriate comparison series.
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05:44Divergence Test (nth Term Test)
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