The binomial series
Problem 7.731 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(1 - 2 x\right)^{\frac{2}{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = -2*x.
- \[ \frac{2 \left(-2\right)^{2} \left(-1 + \frac{2}{3}\right)}{2 \cdot 3} = - \frac{4}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{112 x^{4}}{243} + \frac{32 x^{3}}{81} + \frac{4 x^{2}}{9} + \frac{4 x}{3} + \left(1 - 2 x\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{112 x^{4}}{243} - \frac{32 x^{3}}{81} - \frac{4 x^{2}}{9} - \frac{4 x}{3} + 1 + \cdots \)
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 states the binomial formula and verifies the x² coefficient, but it never actually derives the x, x³, and x⁴ terms. The limit test uses the coefficients from the answer without showing how they were obtained, leaving the student without a clear, correct derivation of the series. This omission misleads the reader into thinking the work is complete when it is not.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic verification of the coefficients and the limit check confirm the result matches the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly identifies the binomial series formula and parameters. The algebraic verification of the coefficients and the limit check confirm the result matches the stated answer.gpt-oss:20b: fail (misleading) 2026-10-23 — The solution states the binomial formula and verifies the x² coefficient, but it never actually derives the x, x³, and x⁴ terms. The limit test uses the coefficients from the answer without showing how they were obtained, leaving the student without a clear, correct derivation of the series. This omission misleads the reader into thinking the work is complete when it is not.qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly identifies the binomial series formula and the substitution parameters. The provided equations verify the coefficients and the asymptotic behavior, confirming the stated answer is correct.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly identifies the binomial series formula and the substitution values. The unchecked equations verify the coefficient calculation and the overall series expansion, which matches the stated answer.gpt-oss:20b: fail (misleading) 2026-10-22 — The solution only verifies the x² coefficient and then claims that the series matches the function up to x⁴ without actually computing or showing the x³ and x⁴ terms. This conclusion is unsupported and could mislead a student into thinking the series has been fully verified.
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 2026-10-23 with SymPy 1.14.0.