The binomial series
Problem 7.636 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(x + 1\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = x.
- \[ - \frac{3 \cdot 1^{2} \left(-3 - 1\right)}{2} = 6 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 15 x^{4} + 10 x^{3} - 6 x^{2} + 3 x - 1 + \frac{1}{\left(x + 1\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 15 x^{4} - 10 x^{3} + 6 x^{2} - 3 x + 1 + \cdots \)
Lines: 2 proved, 1 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 | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | f minus the polynomial at x = 0.01 and 0.001 |
Reviewers
gpt-oss:20b: fail (misleading) — The solution mis‑states the binomial series: it uses the signs of the terms incorrectly (the series should be 1 – 3x + 6x² – 10x³ + 15x⁴ + ⋯). The limit test also subtracts the wrong polynomial, so a student would learn an incorrect expansion.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for the terms (it lists alternating signs starting with positive for x^4, but the expansion of (1+x)^-3 has al
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for the terms (it lists alternating signs starting with positive for x^4, but the expansion of (1+x)^-3 has algpt-oss:20b: fail (misleading) 2026-10-18 — The solution mis‑states the binomial series: it uses the signs of the terms incorrectly (the series should be 1 – 3x + 6x² – 10x³ + 15x⁴ + ⋯). The limit test also subtracts the wrong polynomial, so a student would learn an incorrect expansion.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+x)^{-3} is 1 - 3x + 6x^2 - 10x^3 + 15x^4gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+x)^{-3} is 1 - 3x + 6x^2 - 10x^3 + 15x^4gpt-oss:20b: pass 2026-10-14
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/binomial_series, checked 2026-10-18 with SymPy 1.14.0.