Geometric series
Problem 7.15 · easy Video
Does \( \displaystyle \sum_{n=1}^{\infty} 4 \) converge? If so, find its sum.
- \[ 4 \]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. 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: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly models the constant series as a geometric series with ratio r=1. While the conclusion (divergence) is correct, the reasoning is flawed because a constant series is not typically analyzed via the geometric series ratio test in this manner; it diverges by the Divergence Test (nth term test) since the limit of the terms is 4, not 0. Identifying r=1 for a constant series is technically valid for a geometric series definition, but stating |r|=1 >= 1 implies the standard geometric series divergence condition, which is a valid argument, however, the setup is misleading because the series is trivially divergent by the nth term test, and labeling it primarily as a geometric series with r=1 is an unnecessary and potentially confusing abstraction for a constant series. More importantly, the prompt asks to find what is wrong. The series is sum 4. This is an arithmetic series with d=0? No, it's just a constant. The geometric series formula sum = a/(1-r) fails at r=1. The test used is correct for geometric series. Is it wrong? A constant series IS a geometric series with r=1. The condition for convergence of a geometric series is |r| < 1. Since |1| is not < 1, it diverges. This logic is actually sound. Wait. Let me re-read the adversarial instruction. 'Find what is wrong'. Is there a subtle error? The problem is sum 4. The solution says r=1. |r|=1 >= 1. Diverges. This is correct. Why would it fail? Perhaps the 'style' or 'misleading' aspect? Or is there an error? The series is sum_{n=1}^inf 4. This is 4 + 4 + 4... This is a geometric series with a=4, r=1. The geometric series converges iff |r|<1. Here |r|=1, so it diverges. The logic holds. However, usually, constant series are handled by the nth term test. Using the geometric series test is valid but unusual. Is it 'wrong'? No. Is it 'misleading'? Maybe. But let's look closer. '4, 4 -- The first term, a.' '1, 1 -- The common ratio, r.' The notation is weird. '4, 4' likely means a_1=4, a_2=4. '1, 1' likely means r=1. The sentence says '|r|=1 >= 1'. This is true. 'so the series diverges'. This follows. Is there a trap? The problem asks 'Does it converge? If so, find its sum.' The answer is 'diverges'. The solution supports this. I will mark it as pass unless I find a strict error. Actually, many textbooks define geometric series as sum a*r^(n-1). For r=1, sum a. This diverges if a!=0. The test |r|<1 is necessary and sufficient. So |r|>=1 implies divergence. This is correct. I will mark it pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly models the constant series as a geometric series with ratio r=1. While the conclusion (divergence) is correct, the reasoning is flawed because a constant series is not typically analyzed via the geometric series ratio test in this manner; it diverges by the Divergence Test (nth term test) since the limit of the terms is 4, not 0. Identifying r=1 for a constant series is technically valid for a geometric series definition, but stating |r|=1 >= 1 implies the standard geometric series divergence condition, which is a valid argument, however, the setup is misleading because the series is trivially divergent by the nth term test, and labeling it primarily as a geometric series with r=1 is an unnecessary and potentially confusing abstraction for a constant series. More importantly, the prompt asks to find what is wrong. The series is sum 4. This is an arithmetic series with d=0? No, it's just a constant. The geometric series formula sum = a/(1-r) fails at r=1. The test used is correct for geometric series. Is it wrong? A constant series IS a geometric series with r=1. The condition for convergence of a geometric series is |r| < 1. Since |1| is not < 1, it diverges. This logic is actually sound. Wait. Let me re-read the adversarial instruction. 'Find what is wrong'. Is there a subtle error? The problem is sum 4. The solution says r=1. |r|=1 >= 1. Diverges. This is correct. Why would it fail? Perhaps the 'style' or 'misleading' aspect? Or is there an error? The series is sum_{n=1}^inf 4. This is 4 + 4 + 4... This is a geometric series with a=4, r=1. The geometric series converges iff |r|<1. Here |r|=1, so it diverges. The logic holds. However, usually, constant series are handled by the nth term test. Using the geometric series test is valid but unusual. Is it 'wrong'? No. Is it 'misleading'? Maybe. But let's look closer. '4, 4 -- The first term, a.' '1, 1 -- The common ratio, r.' The notation is weird. '4, 4' likely means a_1=4, a_2=4. '1, 1' likely means r=1. The sentence says '|r|=1 >= 1'. This is true. 'so the series diverges'. This follows. Is there a trap? The problem asks 'Does it converge? If so, find its sum.' The answer is 'diverges'. The solution supports this. I will mark it as pass unless I find a strict error. Actually, many textbooks define geometric series as sum a*r^(n-1). For r=1, sum a. This diverges if a!=0. The test |r|<1 is necessary and sufficient. So |r|>=1 implies divergence. This is correct. I will mark it pass.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly models the series as geometric with ratio r=1, whereas the terms are constant (4, 4, ...), implying a ratio of 1 but the stagpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly identifies the series as geometric with ratio r=1. The series is a constant series (sum of 4s), which diverges by the Divergence Test (limit of terms is 4, not 0), not by the geometric series ratio test.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.