The binomial series
Problem 7.631 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(2 x + 1\right)^{\frac{2}{3}} \).
- (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.Reviewed
- \[ \frac{2 \cdot 2^{2} \left(-1 + \frac{2}{3}\right)}{2 \cdot 3} = - \frac{4}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{112 x^{4}}{243} - \frac{32 x^{3}}{81} + \frac{4 x^{2}}{9} - \frac{4 x}{3} + \left(2 x + 1\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]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 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 series formula and the substitution u = 2x. The provided equations verify the coefficient of x^2 and the overall agreement of the series with the function up to order x^4, which is sufficient to validate the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the binomial series formula and the substitution u = 2x. The provided equations verify the coefficient of x^2 and the overall agreement of the series with the function up to order x^4, which is sufficient to validate the stated answer.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and the substitution u = 2x. The provided checks confirm the coefficients for x^2 and the aggregate error for terms up to x^4, which is sufficient to verify the first five terms.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution calculates the coefficients for the expansion of (1+u)^r but fails to substitute u=2x into the terms, resulting in incorrect coefficients (e.g., the x^2 term is calculated as -4/9 instead of the correct -16/9). Consequently, the derived series does not 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 2026-10-18 with SymPy 1.14.0.