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Geometric series

Problem 7.1 · easy Video

Does \( \displaystyle \sum_{n=0}^{\infty} - 5 \left(-1\right)^{n} \) converge? If so, find its sum.

\[ -5 \]
The first term, a.
\[ -1 \]
The common ratio, r.
|r| = 1 ≥ 1, so the series diverges.

Answer \( \text{diverges} \)

Bars show the terms -5(-1)^n for n = 0 to 23; dots show the running totals. The totals never settle: the series diverges.

Lines: 3 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. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
2	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 common ratio is incorrectly identified as -1; the terms are constant (-5), so the ratio is 1. While the conclusion of divergence is correct, the reasoning is flawed because |r|=1 is not the only reason for divergence (the terms do not approach 0), and identifying r=-1 is factually wrong for this sequence.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The common ratio is incorrectly identified as -1; the terms are constant (-5), so the ratio is 1. While the conclusion of divergence is correct, the reasoning is flawed because |r|=1 is not the only reason for divergence (the terms do not approach 0), and identifying r=-1 is factually wrong for this sequence.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The common ratio is calculated incorrectly as -1 instead of -1 (wait, (-1)^n alternates 1, -1, 1, -1... so the ratio is -1). Let's re-read carefully
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: inconclusive 2026-09-26 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The common ratio is calculated as -1, but the divergence test for geometric series requires |r| >= 1. While the conclusion is correct, the reasoning
gpt-oss:20b: fail (error) 2026-09-26 — The solution incorrectly applies the ratio test: |r|=1 does not imply divergence; the test is inconclusive. The series actually diverges because its terms do not tend to zero, but this reasoning is missing.

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 2026-10-13 with SymPy 1.14.0.