The binomial series
Problem 7.718 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(2 x + 1\right)^{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = 2*x.
- \[ - \frac{2 \cdot 2^{2} \left(-2 - 1\right)}{2} = 12 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 80 x^{4} + 32 x^{3} - 12 x^{2} + 4 x - 1 + \frac{1}{\left(2 x + 1\right)^{2}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 80 x^{4} - 32 x^{3} + 12 x^{2} - 4 x + 1 + \cdots \)
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: passqwen3.6:27b-mlx: fail (error) — The stated answer has the wrong signs for the terms involving x (it should be 1 - 4x + 12x^2 - 32x^3 + 80x^4). The solution setup is correct, but the final answer provided in the problem statement is incorrect, and the solution does not explicitly derive the correct signs, instead relying on a limit check that confirms the magnitude but not the sign pattern relative to the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The stated answer has the wrong signs for the terms involving x (it should be 1 - 4x + 12x^2 - 32x^3 + 80x^4). The solution setup is correct, but the final answer provided in the problem statement is incorrect, and the solution does not explicitly derive the correct signs, instead relying on a limit check that confirms the magnitude but not the sign pattern relative to the stated answer.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer lists the terms in descending order of power (80x^4 - ... + 1), which is not the standard form for a binomial series expansion (which is a power series in ascending powers of x). Furthermore, the solution does not explicitly derive or list the first five terms as requested, but rather verifies the polynomial via a limit check, failing to provide the actual series expansion steps.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has the wrong signs for the terms. The binomial expansion of (1+2x)^{-2} yields 1 - 4x + 12x^2 - 32x^3 + 80x^4, but the stated ansgpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.