The binomial series
Problem 7.713 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(x + 1\right)^{\frac{3}{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = x.Reviewed
- \[ \frac{3 \cdot 1^{2} \left(-1 + \frac{3}{2}\right)}{2 \cdot 2} = \frac{3}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{3 x^{4}}{128} + \frac{x^{3}}{16} - \frac{3 x^{2}}{8} - \frac{3 x}{2} + \left(x + 1\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{3 x^{4}}{128} - \frac{x^{3}}{16} + \frac{3 x^{2}}{8} + \frac{3 x}{2} + 1 + \cdots \)
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 verifies the coefficients and convergence order. The setup is sound and the conclusion follows.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the binomial series formula and verifies the coefficients and convergence order. The setup is sound and the conclusion follows.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial series formula and parameters. The algebraic verification confirms the coefficients match the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer contains incorrect signs for the odd-power terms (x and x^3). The binomial expansion of (1+x)^(3/2) yields positive coefficients for all terms because the factors in the numerator (3/2, 1/2, -1/2, -3/2) result in positive products when divided by the factorials. Specifically, the x term is (3/2)x and the x^3 term is (1/16)x^3, but the stated answer has -3x/2 and -x^3/16.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution only verifies the x² coefficient and then claims agreement up to x⁴ without actually computing or showing the higher‑order terms. This could mislead a student into thinking the series is fully derived when it is not.
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 2026-10-19 with SymPy 1.14.0.