Geometric series
Problem 7.74 · easy Video
Does \( \displaystyle \sum_{n=1}^{\infty} - 3 \left(-1\right)^{n - 1} \) converge? If so, find its sum.
- \[ -3 \]The first term, a.Reviewed
- \[ -1 \]The common ratio, r.Reviewed
- |r| = 1 ≥ 1, so the series diverges.Reviewed
Answer \( \text{diverges} \)
✓ Nihil obstat Lines: 3 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 3 | 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 | the terms do not shrink (60th term checked) |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (style) 2026-09-27 — [domain objection, downgraded to style] The common ratio is calculated incorrectly; for the series -3(-1)^(n-1), the ratio is -1, but the solution lists 'r' as -1, -1 which is confusing notation, and more importantly, the divergence test for geometric series requires |r| >= 1. While |r|=1 does imply divergence, the standard geometric series formula sum = a/(1-r) is undefined here, but the series oscillates (-3, 3, -3, 3...) and does not converge. The logic is actually correct that it diverges, but the notation in step 2 is sloppy. Wait, let me re-read. Step 2 says '-1, -1 -- The common ratio, r.' This is likely a typo for just '-1'. Step 3 says '|r| = 1 >= 1, so the series diverges.' This is a correct application of the geometric series divergence test. However, is the series actually geometric? a_n = -3(-1)^(n-1). a_1 = -3, a_2 = 3, a_3 = -3. Ratio is -1. Yes. Does it diverge? Yes, the terms do not approach 0. The conclusion is correct. Why would it fail? Let's look closer. Is there a subtle error? No, the series diverges. The solution is correct. Wait, the prompt asks me to find what is WRONG. If it's correct, I should pass. But usually these adversarial reviews have a trap. Let's look at the bounds or setup. 'Does ... converge?'. Answer: diverges. Solution: Geometric series with r=-1. |r|=1. Diverges. This is standard. Is there a nuance? Some texts say geometric series converge if |r|<1 and diverge if |r|>=1. This is correct. Maybe the notation in step 2 is the issue? '-1, -1' is not a standard way to write a number. It looks like a list. This is a style error or a minor error. But is it 'wrong'? It's ambiguous. Let's look at the 'Stated answer: diverges'. The solution concludes diverges. It matches. I will mark it as pass because the math is sound, despite the weird notation in step 2 which is likely a parsing artifact of the 'proved' equation line. Actually, looking at step 1: '-3, -3'. Step 2: '-1, -1'. This looks like the CAS outputted a list or vector. The sentence interpretation is 'The first term is -3' and 'The common ratio is -1'. This is correct. I will pass.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the series as geometric with |r|=1 and applies the divergence criterion for geometric series. The conclusion is correct.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.