The binomial series
Problem 7.716 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(\frac{x}{2} + 1\right)^{\frac{3}{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = x/2.
- \[ \left(1 \cdot \frac{1}{2}\right)^{2} \cdot 3 \left(-1 + \frac{3}{2}\right) \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{32} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{3 x^{4}}{2048} + \frac{x^{3}}{128} - \frac{3 x^{2}}{32} - \frac{3 x}{4} + \left(\frac{x}{2} + 1\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{3 x^{4}}{2048} - \frac{x^{3}}{128} + \frac{3 x^{2}}{32} + \frac{3 x}{4} + 1 + \cdots \)
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 (error) — The solution does not actually list the first five terms of the series; it only verifies that the first four terms match the function. The required terms are missing, so the answer is incomplete.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial expansion formula and the substitution variables. The algebraic checks confirm the coefficients match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the binomial expansion formula and the substitution variables. The algebraic checks confirm the coefficients match the stated answer.gpt-oss:20b: fail (error) 2026-10-19 — The solution does not actually list the first five terms of the series; it only verifies that the first four terms match the function. The required terms are missing, so the answer is incomplete.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution claims to find the first five terms but only explicitly calculates the coefficient for the x² term in line 2. It fails to show the derivation or calculation for the x³ and x⁴ terms, which are part of the required five terms.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer contains incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+x/2)^{3/2}$ yields positive coefficients for all terms, but the stated answer has negative coefficients for the cubic and quartic terms. The solution's verification step (line 3) is also flawed because it checks against the incorrect stated answer rather than the true expansion, effectively confirming the error.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution states that the terms agree with the function to order x⁴, but it only computes the x² coefficient and does not actually list the first five terms of the series. This omission misleads a reader into thinking the expansion has been fully derived.
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.