The binomial series
Problem 7.592 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{1 - 2 x} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/3 and u = -2*x.
- \[ \frac{\left(-2\right)^{2} \left(-1 + 1 \cdot \frac{1}{3}\right)}{2 \cdot 3} = - \frac{4}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{160 x^{4}}{243} + \frac{40 x^{3}}{81} + \frac{4 x^{2}}{9} + \frac{2 x}{3} + \sqrt[3]{1 - 2 x} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{160 x^{4}}{243} - \frac{40 x^{3}}{81} - \frac{4 x^{2}}{9} - \frac{2 x}{3} + 1 + \cdots \)
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 that the series terms agree with the function to order x⁴, but the signs of the computed coefficients are wrong (they should be negative). The limit test is also mis‑applied because the series terms are entered with the wrong signs, so a student would learn an incorrect series expansion.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic verification of the coefficient and the limit check for the series expansion are accurate and sufficient to confirm the first five terms.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial series formula and parameters. The algebraic verification of the coefficient and the limit check for the series expansion are accurate and sufficient to confirm the first five terms.gpt-oss:20b: fail (misleading) 2026-10-17 — The solution states that the series terms agree with the function to order x⁴, but the signs of the computed coefficients are wrong (they should be negative). The limit test is also mis‑applied because the series terms are entered with the wrong signs, so a student would learn an incorrect series expansion.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer, and the limit check verifies the expansion is correct up to the fourth order.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer.gpt-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 2026-10-17 with SymPy 1.14.0.