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


∑ (k = 1 to ∞) 20 / (∛k + √k)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{20}{\sqrt[3]{k} + \sqrt{k}} \). We want to determine if this series converges or diverges.
2. Analyze the behavior of the terms for large \( k \). Notice that \( \sqrt[3]{k} = k^{1/3} \) and \( \sqrt{k} = k^{1/2} \). Since \( k^{1/2} \) grows faster than \( k^{1/3} \), the denominator behaves roughly like \( k^{1/2} \) for large \( k \).
3. Simplify the general term for large \( k \) to compare it with a simpler series. The term behaves like \( \frac{20}{k^{1/2}} \) because \( \sqrt{k} \) dominates \( \sqrt[3]{k} \). So, consider the comparison series \( \sum_{k=1}^{\infty} \frac{1}{k^{1/2}} \).
4. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \). Here, \( p = \frac{1}{2} < 1 \), so the comparison series diverges.
5. Use the Comparison Test or Limit Comparison Test: Since the original terms behave like \( \frac{1}{k^{1/2}} \) and this comparison series diverges, conclude that the original series also diverges.

Key Concepts

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 Involving Roots

Understanding how terms with roots like cube roots and square roots behave as the index grows large is crucial. For example, ∛k grows slower than √k, so the dominant term in the denominator affects the term's size. Analyzing this helps in choosing an appropriate comparison series.