Justifying a Series Test Choice
A student must determine whether three series converge, and she has a habit of reaching for the Ratio Test every time. The series are: (A) Σ (n!)/(2^n) from n = 1 to ∞ (B) Σ 1/(n² + 3n) from n = 1 to ∞ (C) Σ (−1)^n /√(n+1) from n = 1 to ∞ Her study partner insists the Ratio Test settles all three because 'it's the most powerful one.' The student suspects each series calls for a different tool and wants to match the right test to each, including checking the hypotheses each test requires before applying it.
1. Apply the Ratio Test to series (A) and determine convergence or divergence, showing the limit computation.
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For (A), aₙ = n!/2ⁿ, all terms positive.
aₙ₊₁/aₙ = [(n+1)!/2^(n+1)] · [2ⁿ/n!] = (n+1)/2.
lim(n→∞) (n+1)/2 = ∞ > 1.
By the Ratio Test, since the limit exceeds 1 (indeed diverges to infinity), series (A) diverges. (Equivalently, terms grow without bound, so the nth-term test also confirms divergence.) The Ratio Test is the natural choice here because factorials and exponentials cancel cleanly.
2. For series (B), explain why the Ratio Test is a poor choice, then determine convergence with an appropriate test. For series (C), state which test applies and whether convergence is conditional or absolute.
Read the worked answer
(B): The Ratio Test limit is lim (aₙ₊₁/aₙ) = lim (n²+3n)/((n+1)²+3(n+1)) = 1, which is inconclusive. A comparison test is better. Since 1/(n²+3n) < 1/n² and Σ1/n² is a convergent p-series (p = 2 > 1), the Direct Comparison Test gives convergence. (Limit Comparison with 1/n² also works: the ratio → 1.) The partner's 'most powerful' framing is misleading - a test that returns 1 has told you nothing.
(C): This is an alternating series with bₙ = 1/√(n+1). bₙ is positive, decreasing (denominator grows), and lim bₙ = 0, so the Alternating Series Test gives convergence. For absolute convergence, consider Σ 1/√(n+1), a p-series with p = 1/2 ≤ 1, which diverges. Therefore (C) converges conditionally, not absolutely.
Each series needed a different test; hypotheses (positivity, monotone decreasing, limit zero) were checked before applying.