Home›Calculus 2›The binomial series›Problem 7.598

The binomial series

Problem 7.598 · medium

Find the first five terms of the binomial series for \( \displaystyle \left(1 - x\right)^{\frac{2}{3}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = -x.
\[ \frac{2 \left(-1\right)^{2} \left(-1 + \frac{2}{3}\right)}{2 \cdot 3} = - \frac{1}{9} \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{\frac{7 x^{4}}{243} + \frac{4 x^{3}}{81} + \frac{x^{2}}{9} + \frac{2 x}{3} + \left(1 - x\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution only verifies the coefficient of \(x^2\) and checks agreement up to \(x^4\); it does not actually list the first five terms of the series, so
qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The provided equations verify the 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 coefficients and the overall expansion accuracy, matching the stated answer.
gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution only verifies the coefficient of \(x^2\) and checks agreement up to \(x^4\); it does not actually list the first five terms of the series, so
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer.
gpt-oss:20b: fail (misleading) 2026-10-15 — The solution only verifies the x² coefficient and a limit check; it does not actually list the first five terms of the series, so a student would not learn the full expansion.
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the binomial series formula and parameters. The unchecked equations verify the coefficient and the overall expansion accuracy.
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.