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Absolute and conditional convergence

Problem 7.243 · easy

Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n + 1} \cdot 2^{- n} 3^{n} \) converge absolutely, converge conditionally, or diverge?

|aₙ| grows without bound, so the terms do not go to 0 (divergence test): the series diverges.
Reviewed

Answer \( \text{diverges} \)

✓ Nihil obstat Lines: 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	SymPy's Sum.is_convergent on the series and on its absolute values

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies that the magnitude of the terms |a_n| = (3/2)^n grows without bound, implying the terms do not approach zero. This correctly applies the Divergence Test to conclude the series diverges.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that the magnitude of the terms |a_n| = (3/2)^n grows without bound, implying the terms do not approach zero. This correctly applies the Divergence Test to conclude the series diverges.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that the magnitude of the terms |a_n| = (3/2)^n grows without bound, implying the terms do not approach zero. This correctly applies the Divergence Test to conclude divergence.
gpt-oss:20b: pass 2026-10-04

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/alternating_series, checked 2026-10-04 with SymPy 1.14.0.