The binomial series
Problem 7.658 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(3 x + 1\right)^{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = 3*x.
- \[ - \frac{2 \cdot 3^{2} \left(-2 - 1\right)}{2} = 27 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 405 x^{4} + 108 x^{3} - 27 x^{2} + 6 x - 1 + \frac{1}{\left(3 x + 1\right)^{2}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 405 x^{4} - 108 x^{3} + 27 x^{2} - 6 x + 1 + \cdots \)
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 verifies some coefficients but never actually lists the first five terms of the series, so it does not fully answer the problem as stated.qwen3.6:27b-mlx: fail (error) — The stated answer has incorrect signs for the terms. Since r = -2, the coefficients are positive (e.g., r(r-1)/2 = 3), but the stated answer lists negative coefficients for x, x^3, and x^4 (e.g., -6x instead of 6x). The solution setup is correct, but the final answer provided in the problem statement is wrong, and the solution does not explicitly derive the correct signs to contradict the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer has incorrect signs for the terms. Since r = -2, the coefficients are positive (e.g., r(r-1)/2 = 3), but the stated answer lists negative coefficients for x, x^3, and x^4 (e.g., -6x instead of 6x). The solution setup is correct, but the final answer provided in the problem statement is wrong, and the solution does not explicitly derive the correct signs to contradict the stated answer.gpt-oss:20b: fail (style) 2026-10-18 — The solution verifies some coefficients but never actually lists the first five terms of the series, so it does not fully answer the problem as stated.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer has the wrong signs for the terms (it lists alternating signs starting with positive for x^4, but the expansion of (1+3x)^-2 has all negative coefficients for x^n where n>=1). The solution's limit check confirms the correct series is 1 - 6x + 27x^2 - 108x^3 + 405x^4, which contradicts the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-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. The binomial expansion of (1+3x)^{-2} is 1 - 6x + 27x^2 - 108x^3 + 405x^4 - ..., bugpt-oss:20b: pass 2026-10-15
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.