The binomial series
Problem 7.657 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{2 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = 2*x.Reviewed
- \[ \frac{2^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = \frac{8}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{560 x^{4}}{243} + \frac{112 x^{3}}{81} - \frac{8 x^{2}}{9} + \frac{2 x}{3} - 1 + \frac{1}{\sqrt[3]{2 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{560 x^{4}}{243} - \frac{112 x^{3}}{81} + \frac{8 x^{2}}{9} - \frac{2 x}{3} + 1 + \cdots \)
Lines: 2 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. Although the stated answer lists terms in descending order of power, the set of terms is correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. Although the stated answer lists terms in descending order of power, the set of terms is correct.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 the wrong signs for the odd-powered terms. The binomial expansion of (1+2x)^{-1/3} yields a negative coefficient for x (specifigpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15gpt-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.