The binomial series
Problem 7.630 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(1 - 2 x\right)^{\frac{3}{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = -2*x.
- \[ \frac{3 \left(-2\right)^{2} \left(-1 + \frac{3}{2}\right)}{2 \cdot 2} = \frac{3}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{3 x^{4}}{8} - \frac{x^{3}}{2} - \frac{3 x^{2}}{2} + 3 x + \left(1 - 2 x\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{3 x^{4}}{8} + \frac{x^{3}}{2} + \frac{3 x^{2}}{2} - 3 x + 1 + \cdots \)
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 (style) — The solution states the binomial formula and verifies one coefficient, but it never actually writes out the first five terms of the series. The limit check is correct, but the missing explicit terms make the answer incomplete and unclear for a student.qwen3.6:27b-mlx: fail (error) — The stated answer lists the terms in descending order of power (x^4 down to constant), but the coefficients are incorrect for the higher powers. Specifically, the coefficient for x^3 is given as 1/2, but the correct binomial coefficient calculation yields -1. The coefficient for x^4 is given as 3/8, but the correct value is -1/2. The solution claims the terms agree to order x^4, but the provided polynomial does not match the expansion of (1-2x)^(3/2).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer lists the terms in descending order of power (x^4 down to constant), but the coefficients are incorrect for the higher powers. Specifically, the coefficient for x^3 is given as 1/2, but the correct binomial coefficient calculation yields -1. The coefficient for x^4 is given as 3/8, but the correct value is -1/2. The solution claims the terms agree to order x^4, but the provided polynomial does not match the expansion of (1-2x)^(3/2).gpt-oss:20b: fail (style) 2026-10-17 — The solution states the binomial formula and verifies one coefficient, but it never actually writes out the first five terms of the series. The limit check is correct, but the missing explicit terms make the answer incomplete and unclear for a student.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer contains incorrect signs for the odd-powered terms (x and x^3). The correct expansion is 1 - 3x + (3/2)x^2 - (1/2)x^3 + (3/8)x^4, but the answer lists +3x and +x^3. The solution's verification step (line 3) is also flawed because it checks against the incorrect stated answer rather than the true function expansion.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims the coefficient of x^3 is 1/2, but the correct coefficient is -1/2. The binomial expansion yields 1 - 3x + (3/2)x^2 - (1/2)x^3 + (3/8)x^4. The stated answer has the wrong sign for the x^3 term, and the limit check in line 3 is likely flawed or misinterpreted because it does not account for the sign error in the proposed series.gpt-oss:20b: fail (style) 2026-10-14 — The solution does not explicitly state the first five terms of the series, leaving the answer incomplete.
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.