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)

검증된 단계별 안내

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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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 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 terms behave similarly for large indices.

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.

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)