The binomial series
Problem 7.717 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{3 x + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/3 and u = 3*x.
- \[ \frac{3^{2} \left(-1 + 1 \cdot \frac{1}{3}\right)}{2 \cdot 3} = -1 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{10 x^{4}}{3} - \frac{5 x^{3}}{3} + x^{2} - x + \sqrt[3]{3 x + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{10 x^{4}}{3} + \frac{5 x^{3}}{3} - x^{2} + x + 1 + \cdots \)
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: fail (error) — The stated answer has incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{1/3}$ yields positive coefficients for all powers of $x$, but the stated answer lists them as negative. Additionally, the solution skips the calculation of the $x^3$ and $x^4$ terms entirely, providing no evidence for the coefficients in the final answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer has incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{1/3}$ yields positive coefficients for all powers of $x$, but the stated answer lists them as negative. Additionally, the solution skips the calculation of the $x^3$ and $x^4$ terms entirely, providing no evidence for the coefficients in the final answer.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer returned a non-objectgpt-oss:20b: fail (misleading) 2026-10-16 — The solution verifies the coefficients but never actually lists the first five terms of the series, leaving the answer incomplete and potentially misleading to a student who expects the explicit expansion.qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. The stated answer matches the derived terms.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.