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Geometric series

Problem 7.69 · easy Video

Does \( \displaystyle \sum_{n=0}^{\infty} - 2 \left(-1\right)^{n} \) converge? If so, find its sum.

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

Answer \( \text{diverges} \)

Bars show the terms -2(-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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The ratio test is inconclusive when |r|=1; it does not guarantee divergence. The series \sum_{n=0}^\infty -2(-1)^n does not converge, but the justifi
qwen3.6:27b-mlx: pass — The solution correctly identifies the series as geometric with |r|=1 and applies the standard divergence criterion for geometric series.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the series as geometric with |r|=1 and applies the standard divergence criterion for geometric series.
gpt-oss:20b: inconclusive 2026-09-28 — reviewer response could not be parsed: {"verdict":"fail","severity":"misleading","notes":"The ratio test is inconclusive when |r|=1; it does not guarantee divergence. The series \sum_{n=0}^\infty -2(-1)^n does not converge, but the justifi
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution incorrectly identifies the common ratio as -1. The terms are 2, 2, 2, ... 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 here; the terms do not approach zero.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the series as geometric with ratio r = -1. Since |r| = 1, the geometric series test correctly concludes divergence. The reasoning is sound and complete.
gpt-oss:20b: pass 2026-09-26

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.