The binomial series
Problem 7.635 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(3 x + 1\right)^{\frac{3}{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = 3*x.
- \[ \frac{3 \cdot 3^{2} \left(-1 + \frac{3}{2}\right)}{2 \cdot 2} = \frac{27}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{243 x^{4}}{128} + \frac{27 x^{3}}{16} - \frac{27 x^{2}}{8} - \frac{9 x}{2} + \left(3 x + 1\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{243 x^{4}}{128} - \frac{27 x^{3}}{16} + \frac{27 x^{2}}{8} + \frac{9 x}{2} + 1 + \cdots \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the coefficients for the odd powers of x. The term for x^1 should be 9/2 x, but the stated answer has +9x/2 (corr
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly calculates the coefficients for the odd powers of x. The term for x^1 should be 9/2 x, but the stated answer has +9x/2 (corrgpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer contains incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{3/2}$ yields positive coefficients for all terms since $r=3/2 > 0$ and the powers of $u=3x$ alternate in sign in the general formula but are multiplied by alternating signs from the falling factorial, resulting in all positive terms for this specific $r$. Specifically, the coefficient for $x^3$ should be positive ($27/16$) and for $x^4$ positive ($243/128$), but the stated answer has them negative.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer contains incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{3/2}$ yields positive coefficients forgpt-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 2026-10-18 with SymPy 1.14.0.