The binomial series
Problem 7.591 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(2 x + 1\right)^{\frac{3}{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = 2*x.
- \[ \frac{3 \cdot 2^{2} \left(-1 + \frac{3}{2}\right)}{2 \cdot 2} = \frac{3}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{3 x^{4}}{8} + \frac{x^{3}}{2} - \frac{3 x^{2}}{2} - 3 x + \left(2 x + 1\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{3 x^{4}}{8} - \frac{x^{3}}{2} + \frac{3 x^{2}}{2} + 3 x + 1 + \cdots \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 that the terms agree to order x⁴ but never actually lists the first five terms or computes the coefficients for x, x², x³, x⁴. The limit test is not sufficient to claim the series expansion without explicitly showing the terms, so a student would be misled into thinking the answer is complete when it is not.qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the $x^3$, $x^2$, and $x^4$ terms. The binomial expansion of $(1+2x)^{3/2}$ yields positive coeffici
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the $x^3$, $x^2$, and $x^4$ terms. The binomial expansion of $(1+2x)^{3/2}$ yields positive coefficigpt-oss:20b: fail (misleading) 2026-10-17 — The solution states that the terms agree to order x⁴ but never actually lists the first five terms or computes the coefficients for x, x², x³, x⁴. The limit test is not sufficient to claim the series expansion without explicitly showing the terms, so a student would be misled into thinking the answer is complete when it is not.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the odd-powered terms (x and x^3). The binomial expansion of (1+2x)^(3/2) yields positive coefficients forgpt-oss:20b: fail (misleading) 2026-10-15 — The solution verifies the series up to order x⁴ but presents the polynomial with incorrect signs, which could mislead a student into thinking the coefficients are negative. The setup and coefficient calculation are correct, but the concluding check is misleading.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer contains incorrect signs for the $x^3$, $x^2$, and $x^4$ terms. The binomial expansion of $(1+2x)^{3/2}$ yields positive coefficients for all terms, but the answer lists negative coefficients for the higher-order terms (e.g., $-x^3/2$ instead of $+x^3/2$). Additionally, the limit check in line 3 uses a polynomial with incorrect signs, so it does not actually verify the stated answer.gpt-oss:20b: fail (error) 2026-10-14 — The solution does not actually compute or list the first five terms of the series; it only states the general binomial expansion formula and gives a check, but omits the explicit coefficients. This omission means the setup is incomplete and a student would not learn how to obtain the required terms.
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.