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Geometric series

Problem 7.222 · easy

Does \( \displaystyle \sum_{n=1}^{\infty} 2 \) converge? If so, find its sum.

\[ 2 \]
The first term, a.✓ Proved
\[ 1 \]
The common ratio, r.✓ Proved
|r| = 1 ≥ 1, so the series diverges.

Answer \( \text{diverges} \)

Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the terms do not shrink (60th term checked)

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the series as geometric with ratio r=1. The series is a constant series (arithmetic with difference 0), not geometric. While the conclusion of divergence is correct, the reasoning is fundamentally flawed because the terms do not form a geometric progression.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly identifies the series as geometric with ratio r=1. The series is a constant series (arithmetic with difference 0), not geometric. While the conclusion of divergence is correct, the reasoning is fundamentally flawed because the terms do not form a geometric progression.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution incorrectly identifies the series as geometric with ratio r=1. The series is actually a constant series (arithmetic with difference 0) where the terms do not approach 0, so it diverges by the Divergence Test, not the geometric series test.
gpt-oss:20b: pass 2026-10-03

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/geometric_series, checked 2026-10-03 with SymPy 1.14.0.