9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k² + 4)

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^2 + 4} \). We want to determine if this series converges.
2. Choose a comparison series that is simpler but behaves similarly for large \( k \). Since \( k^2 + 4 \) behaves like \( k^2 \) for large \( k \), consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 \).
3. Recall that the p-series \( \sum \frac{1}{k^p} \) converges if \( p > 1 \). Since \( p = 2 > 1 \), the comparison series \( \sum \frac{1}{k^2} \) converges.
4. Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{\frac{1}{k^2 + 4}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{k^2}{k^2 + 4} \).
5. Evaluate the limit \( L \). If \( L \) is a finite positive number, then both series either both converge or both diverge. Since the comparison series converges, this will imply the original series converges as well.

Key Concepts

Comparison Test

The Comparison Test determines the convergence of a series by comparing it to another 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 series have similar term behavior.

p-Series and Their Convergence

A p-series has the form ∑ 1/k^p and converges if p > 1, diverging otherwise. Recognizing that 1/(k² + 4) behaves like 1/k² for large k helps apply comparison tests effectively, since the p-series with p=2 is known to converge.