The binomial series
Problem 7.584 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{\frac{x}{2} + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/3 and u = x/2.
- \[ \frac{\left(1 \cdot \frac{1}{2}\right)^{2} \left(-1 + 1 \cdot \frac{1}{3}\right)}{2 \cdot 3} = - \frac{1}{36} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{1944} - \frac{5 x^{3}}{648} + \frac{x^{2}}{36} - \frac{x}{6} + \sqrt[3]{\frac{x}{2} + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{5 x^{4}}{1944} + \frac{5 x^{3}}{648} - \frac{x^{2}}{36} + \frac{x}{6} + 1 + \cdots \)
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 (error) — Sentence 3 uses the wrong signs for the x³ and x⁴ terms when checking the series; the test does not correspond to the stated series and therefore does not verify the correct expansion.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial expansion parameters and verifies the resulting polynomial terms against the function. The setup is sound and the verification confirms the coefficients.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial expansion parameters and verifies the resulting polynomial terms against the function. The setup is sound and the verification confirms the coefficients.gpt-oss:20b: fail (error) 2026-10-17 — Sentence 3 uses the wrong signs for the x³ and x⁴ terms when checking the series; the test does not correspond to the stated series and therefore does not verify the correct expansion.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients algebraically and via limits. The setup and conclusion are sound.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer has incorrect signs for the cubic and quartic terms. The binomial expansion of (1 + x/2)^(1/3) yields positive coefficients for x^3 and x^4 (specifically 5/648 and 5/1944), but the stated answer lists them as negative. The solution text claims these terms agree with the function, which is false.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.