The binomial series
Problem 7.593 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(x + 1\right)^{\frac{2}{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = x.
- \[ \frac{2 \cdot 1^{2} \left(-1 + \frac{2}{3}\right)}{2 \cdot 3} = - \frac{1}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{7 x^{4}}{243} - \frac{4 x^{3}}{81} + \frac{x^{2}}{9} - \frac{2 x}{3} + \left(x + 1\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{7 x^{4}}{243} + \frac{4 x^{3}}{81} - \frac{x^{2}}{9} + \frac{2 x}{3} + 1 + \cdots \)
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 (misleading) — The solution states the binomial expansion formula and verifies the x² coefficient, but it never actually lists the first five terms of (x+1)^{2/3}. A student would be left without the requested result, which is misleading.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The provided equations verify the specific coefficients and the overall expansion accuracy, matching the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the binomial series formula and parameters. The provided equations verify the specific coefficients and the overall expansion accuracy, matching the stated answer.gpt-oss:20b: fail (misleading) 2026-10-17 — The solution states the binomial expansion formula and verifies the x² coefficient, but it never actually lists the first five terms of (x+1)^{2/3}. A student would be left without the requested result, which is misleading.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients and the order of accuracy, matching the stated answer.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients 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 2026-10-17 with SymPy 1.14.0.