The binomial series
Problem 7.595 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{3 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = 3*x.
- \[ \frac{3^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = \frac{27}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{2835 x^{4}}{128} + \frac{135 x^{3}}{16} - \frac{27 x^{2}}{8} + \frac{3 x}{2} - 1 + \frac{1}{\sqrt{3 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{2835 x^{4}}{128} - \frac{135 x^{3}}{16} + \frac{27 x^{2}}{8} - \frac{3 x}{2} + 1 + \cdots \)
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: fail (error) — The stated answer has incorrect signs for the terms with odd powers of x. The binomial expansion of (1+u)^(-1/2) alternates in sign, so the x term should be -3x/2 and the x^3 term should be -135x^3/16, but the stated answer lists them as positive. The solution's sentence correctly sets up the formula, but the final stated answer provided in the problem description is mathematically incorrect, and the solution fails to derive or correct it, instead just verifying a wrong polynomial.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer has incorrect signs for the terms with odd powers of x. The binomial expansion of (1+u)^(-1/2) alternates in sign, so the x term should be -3x/2 and the x^3 term should be -135x^3/16, but the stated answer lists them as positive. The solution's sentence correctly sets up the formula, but the final stated answer provided in the problem description is mathematically incorrect, and the solution fails to derive or correct it, instead just verifying a wrong polynomial.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer has incorrect signs for the odd-power terms (it lists -3x/2 and -135x^3/16, but the binomial expansion of (1+3x)^(-1/2) yields +3x/2 and +135x^3/16). The solution claims the terms agree with the function, which is false.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer has incorrect signs for the odd-powered terms (x and x^3). The binomial expansion of (1+u)^{-1/2} has alternating signs, so the terms for u=3x should be -3x/2 and -135x^3/16, but the answer lists them as positive. The solution text claims the terms agree with the function, which is false.gpt-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 2026-10-17 with SymPy 1.14.0.