The binomial series
Problem 7.659 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(2 x + 1\right)^{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = 2*x.
- \[ - \frac{3 \cdot 2^{2} \left(-3 - 1\right)}{2} = 24 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 240 x^{4} + 80 x^{3} - 24 x^{2} + 6 x - 1 + \frac{1}{\left(2 x + 1\right)^{3}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 240 x^{4} - 80 x^{3} + 24 x^{2} - 6 x + 1 + \cdots \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for all terms except the constant. The binomial expansion of (1+2x)^{-3} yields 1 - 6x + 24x^2 - 80x^3 + 240x^
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 all terms except the constant. The binomial expansion of (1+2x)^{-3} yields 1 - 6x + 24x^2 - 80x^3 + 240x^gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the terms involving x, x^3, and x^4. Since u=2x and r=-3, all terms in the expansion of (1+2x)^{-3} are pogpt-oss:20b: fail (error) 2026-10-16 — The solution incorrectly signs the series terms in the limit test (it uses -240x⁴+80x³-24x²+6x-1 instead of 240x⁴-80x³+24x²-6x+1) and does not actually list the first five terms of the binomial expansion as required.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. The binomial expansion of (1+2x)^{-3} yields 1 - 6x + 24x^2 - 80x^3 +gpt-oss:20b: pass 2026-10-15
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.