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Communicating Convergence Results Precisely
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Calculus I
CAL1.168 Communicating Convergence Results Precisely
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Select the option that correctly classifies the series as absolutely convergent, conditionally convergent, or divergent AND gives a valid supporting test statement. Series: Σ (n=1→∞) [n² ÷ (n²+1)] · n⁻¹
A
Divergent by limit comparison with Σ 1÷n: ( (n²÷(n²+1))·(1÷n) ) ÷ (1÷n) → 1.
B
Absolutely convergent because the terms are smaller than a convergent geometric series with ratio 1 ÷ 2.
C
Conditionally convergent: the series alternates and bₙ is decreasing to 0, so it passes the alternating series test.
D
Divergent because it is a p-series with p = 2, which forces divergence.
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