The binomial series
Problem 7.586 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(\frac{x}{2} + 1\right)^{\frac{2}{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = x/2.Reviewed
- \[ \left(1 \cdot \frac{1}{2}\right)^{2} \cdot 2 \left(-1 + \frac{2}{3}\right) \frac{1}{3} \cdot \frac{1}{2} = - \frac{1}{36} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{7 x^{4}}{3888} - \frac{x^{3}}{162} + \frac{x^{2}}{36} - \frac{x}{3} + \left(\frac{x}{2} + 1\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{7 x^{4}}{3888} + \frac{x^{3}}{162} - \frac{x^{2}}{36} + \frac{x}{3} + 1 + \cdots \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 formula and the substitution variables. The provided checks confirm the coefficients and the overall series structure match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial expansion formula and the substitution variables. The provided checks confirm the coefficients and the overall series structure match the stated answer.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial expansion formula and the substitution variables. The provided checks confirm the coefficients for the x^2 term and the overall agreement up to x^4, which validates the first five terms (constant through x^4).gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the binomial series formula and the substitution variables. The unchecked equations confirm the coefficients and the validity of the expansion up to the fourth order.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.