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MathCalculus

Alternating Series Test: Classify ∑ (−1)^(n+1)√(n+3)/n

Verify decreasing magnitudes and a zero limit, then compare the absolute terms with 1/√n to establish conditional convergence.

Question

Classify the following series as absolutely convergent, conditionally convergent, or divergent:

∑n=1∞(−1)n+1n+3n.\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n+3}}{n}.

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Evidence boundary

This solution addresses OpenStax §5.5 exercise 253 with n starting at 1. It classifies convergence; it does not claim an exact sum. The derivative and comparison are independently derived from the stated term.