Geometric series
Problem 7.173 · easy Video
Does \( \displaystyle \sum_{n=1}^{\infty} \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ 1 \]The first term, a.
- \[ -1 \]The common ratio, r.
- |r| = 1 ≥ 1, so the series diverges.
Answer \( \text{diverges} \)
Lines: 3 not checked. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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: passqwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution incorrectly identifies the series as geometric. The terms are 1, -1, 1, -1..., which do not form a geometric progression with a constant ratio (the ratio alternates between -1 and -1, but the standard geometric series test requires a constant r for all n, and more importantly, the series is not defined by a geometric formula a*r^(n-1) in a way that allows summing via the geometric series formula when |r|=1. While the conclusion 'diverges' is correct, the reasoning is flawed because the series is not geometric in the sense required for the ratio test or geometric series sum formula application here; it is an alternating series where the terms do not approach 0, so it diverges by the Divergence Test. Labeling it geometric with r=-1 is technically true for the ratio of consecutive terms, but the justification '|r|=1 >= 1 so it diverges' is a misapplication of the geometric series convergence criteria which usually states divergence for |r|>=1, but the primary error is that this is not a standard geometric series problem setup, and the divergence is more directly shown by the nth term test. However, the most critical error is that the solution implies the geometric series test is the correct tool, whereas the Divergence Test is the fundamental reason. Actually, a geometric series with |r|>=1 does diverge, so the logic holds for a geometric series. But is this a geometric series? Yes, a_n = (-1)^(n-1) = 1 * (-1)^(n-1). So it IS a geometric series with a=1, r=-1. The geometric series sum formula only works for |r|<1. For |r|>=1, it diverges. So the statement is actually correct. Wait. Let me re-read. 'Does sum (-1)^(n-1) converge?' Terms: 1, -1, 1, -1... Partial sums: 1, 0, 1, 0... Oscillates. Diverges. The solution says it's geometric with r=-1. |r|=1. Geometric series diverges if |r|>=1. This is a standard theorem. So the solution is actually correct. Why would it be fail? Maybe the prompt implies I should find it wrong? 'Your job is to find what is wrong'. If it's correct, I say pass. Is there a nuance? The geometric series test is often stated as: If |r|<1, converges. If |r|>=1, diverges. This is true. So the solution is sound. Let me double check if there's any 'misleading' aspect. It's a very simple series. Using the geometric series classification is valid. I will mark it as pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — [domain objection, downgraded to style] The solution incorrectly identifies the series as geometric. The terms are 1, -1, 1, -1..., which do not form a geometric progression with a constant ratio (the ratio alternates between -1 and -1, but the standard geometric series test requires a constant r for all n, and more importantly, the series is not defined by a geometric formula a*r^(n-1) in a way that allows summing via the geometric series formula when |r|=1. While the conclusion 'diverges' is correct, the reasoning is flawed because the series is not geometric in the sense required for the ratio test or geometric series sum formula application here; it is an alternating series where the terms do not approach 0, so it diverges by the Divergence Test. Labeling it geometric with r=-1 is technically true for the ratio of consecutive terms, but the justification '|r|=1 >= 1 so it diverges' is a misapplication of the geometric series convergence criteria which usually states divergence for |r|>=1, but the primary error is that this is not a standard geometric series problem setup, and the divergence is more directly shown by the nth term test. However, the most critical error is that the solution implies the geometric series test is the correct tool, whereas the Divergence Test is the fundamental reason. Actually, a geometric series with |r|>=1 does diverge, so the logic holds for a geometric series. But is this a geometric series? Yes, a_n = (-1)^(n-1) = 1 * (-1)^(n-1). So it IS a geometric series with a=1, r=-1. The geometric series sum formula only works for |r|<1. For |r|>=1, it diverges. So the statement is actually correct. Wait. Let me re-read. 'Does sum (-1)^(n-1) converge?' Terms: 1, -1, 1, -1... Partial sums: 1, 0, 1, 0... Oscillates. Diverges. The solution says it's geometric with r=-1. |r|=1. Geometric series diverges if |r|>=1. This is a standard theorem. So the solution is actually correct. Why would it be fail? Maybe the prompt implies I should find it wrong? 'Your job is to find what is wrong'. If it's correct, I say pass. Is there a nuance? The geometric series test is often stated as: If |r|<1, converges. If |r|>=1, diverges. This is true. So the solution is sound. Let me double check if there's any 'misleading' aspect. It's a very simple series. Using the geometric series classification is valid. I will mark it as pass.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly identifies the series as geometric with ratio r = -1. While the terms alternate between 1 and -1, the ratio test is inconclusive for |r|=1, and the series actually diverges by the Divergence Test (terms do not approach 0), not because it is a divergent geometric series in the standard summation sense.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the series as geometric with ratio r=-1. The terms are 1, -1, 1, -1..., so the ratio is indeed -1, but the divergpt-oss:20b: pass 2026-09-28
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.