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The binomial series

Problem 7.594 · medium

Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(4 x + 1\right)^{3}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -3 and u = 4*x.
\[ - \frac{3 \cdot 4^{2} \left(-3 - 1\right)}{2} = 96 \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 3840 x^{4} + 640 x^{3} - 96 x^{2} + 12 x - 1 + \frac{1}{\left(4 x + 1\right)^{3}}}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( 3840 x^{4} - 640 x^{3} + 96 x^{2} - 12 x + 1 + \cdots \)

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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The coefficient of \(x^2\) is computed incorrectly: \((-3)(-4)/2 \times 4^2 = 192\), not 96. The subsequent series terms also have incorrect signs and ma
qwen3.6:27b-mlx: fail (error) — The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+4x)^{-3} yields alternating signs (1 - 12x + 96x^2 - ...), but the stated answer lists them as positive (except the constant). The solution's sentence correctly sets up the formula, but the final stated answer contradicts the correct application of that formula and the provided check (which uses the correct signs in the limit expression).

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+4x)^{-3} yields alternating signs (1 - 12x + 96x^2 - ...), but the stated answer lists them as positive (except the constant). The solution's sentence correctly sets up the formula, but the final stated answer contradicts the correct application of that formula and the provided check (which uses the correct signs in the limit expression).
gpt-oss:20b: inconclusive 2026-10-17 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The coefficient of \(x^2\) is computed incorrectly: \((-3)(-4)/2 \times 4^2 = 192\), not 96. The subsequent series terms also have incorrect signs and ma
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 all terms except the constant term. The binomial expansion of (1+4x)^{-3} yields alternating signs (1 - 12
gpt-oss:20b: fail (error) 2026-10-15 — The solution’s third sentence uses the series with all signs reversed (−3840x⁴+640x³−96x²+12x−1) instead of the correct 3840x⁴−640x³+96x²−12x+1, so the limit test does not actually verify the stated series. This is a substantive mistake in the setup of the verification step.
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer has incorrect signs for the terms involving x, x^2, x^3, and x^4. Since r = -3 is negative, the binomial expansion of (1+u)^r alternates in sign, but the coefficients r(r-1)... are all negative, making the terms negative for positive u. Specifically, the expansion is 1 - 12x + 96x^2 - 640x^3 + 3840x^4... The stated answer has the signs flipped for all non-constant terms (e.g., -12x vs +12x). Additionally, the solution text does not explicitly derive the full series or explain the sign pattern, relying on unchecked equations that do not match the stated answer's signs.
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.