The binomial series
Problem 7.655 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(4 x + 1\right)^{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = 4*x.
- \[ - \frac{2 \cdot 4^{2} \left(-2 - 1\right)}{2} = 48 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 1280 x^{4} + 256 x^{3} - 48 x^{2} + 8 x - 1 + \frac{1}{\left(4 x + 1\right)^{2}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 1280 x^{4} - 256 x^{3} + 48 x^{2} - 8 x + 1 + \cdots \)
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 incorrect signs for all terms except the constant term. The binomial expansion of (1+4x)^{-2} yields alternating signs starting with positive, i.e., 1 - 8x + 48x^2 - 256x^3 + 1280x^4, but the stated answer lists them as 1 - 8x + 48x^2 - 256x^3 + 1280x^4? Wait, let's re-read carefully. Stated: 1280x^4 - 256x^3 + 48x^2 - 8x + 1. This is actually correct in value (order doesn't matter for a polynomial). However, the solution text is extremely sparse and does not actually derive the terms, just checks one coefficient and the limit. The prompt asks to review the solution. The solution consists of a setup sentence and two checked equations. The setup is correct. The equations are correct. But does the solution *find* the terms? It asserts the formula and then verifies the result. Is this a valid solution? Usually, a solution should show the derivation. However, the instructions say 'Each line... is either an EQUATION... or a SENTENCE'. The solution provided is technically a verification, not a derivation. But more importantly, look at the stated answer vs the problem. Problem: 1/(4x+1)^2. Expansion: 1 - 8x + 48x^2 - 256x^3 + 1280x^4. Stated answer: 1280x^4 - 256x^3 + 48x^2 - 8x + 1. These are identical polynomials. The solution is mathematically sound in its claims. Why would it fail? Let's look closer. Line 1 sets up the formula. Line 2 checks the x^2 coefficient. Line 3 checks the limit of the difference divided by x^4 is 0, implying the polynomial matches up to x^4. This is a valid verification method. Is there an error? The problem asks to 'Find' the terms. The solution 'Verifies' the terms. In many contexts, this is acceptable if the answer is given. But wait, look at the signs in the stated answer again. 1280x^4 (pos) - 256x^3 (neg) + 48x^2 (pos) - 8x (neg) + 1 (pos). The expansion of (1+u)^-2 is 1 - 2u + 3u^2 - 4u^3 + 5u^4. With u=4x: 1 - 8x + 3(16)x^2 - 4(64)x^3 + 5(256)x^4 = 1 - 8x + 48x^2 - 256x^3 + 1280x^4. The stated answer matches. The solution is correct. Why did I think it might fail? Perhaps because it doesn't explicitly list the terms in the text? But the 'Stated answer' is separate. The solution lines support the stated answer. It seems sound. Let me double check the 'adversarial' instruction. 'Find what is wrong'. Is there a subtle issue? Line 1: 'r = -2 and u = 4*x'. Correct. Line 2: Checks x^2 coeff. Correct. Line 3: Checks limit. Correct. The solution is a valid proof that the stated answer is correct. I will mark it pass.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer has incorrect signs for all terms except the constant term. The binomial expansion of (1+4x)^{-2} yields alternating signs starting with positive, i.e., 1 - 8x + 48x^2 - 256x^3 + 1280x^4, but the stated answer lists them as 1 - 8x + 48x^2 - 256x^3 + 1280x^4? Wait, let's re-read carefully. Stated: 1280x^4 - 256x^3 + 48x^2 - 8x + 1. This is actually correct in value (order doesn't matter for a polynomial). However, the solution text is extremely sparse and does not actually derive the terms, just checks one coefficient and the limit. The prompt asks to review the solution. The solution consists of a setup sentence and two checked equations. The setup is correct. The equations are correct. But does the solution *find* the terms? It asserts the formula and then verifies the result. Is this a valid solution? Usually, a solution should show the derivation. However, the instructions say 'Each line... is either an EQUATION... or a SENTENCE'. The solution provided is technically a verification, not a derivation. But more importantly, look at the stated answer vs the problem. Problem: 1/(4x+1)^2. Expansion: 1 - 8x + 48x^2 - 256x^3 + 1280x^4. Stated answer: 1280x^4 - 256x^3 + 48x^2 - 8x + 1. These are identical polynomials. The solution is mathematically sound in its claims. Why would it fail? Let's look closer. Line 1 sets up the formula. Line 2 checks the x^2 coefficient. Line 3 checks the limit of the difference divided by x^4 is 0, implying the polynomial matches up to x^4. This is a valid verification method. Is there an error? The problem asks to 'Find' the terms. The solution 'Verifies' the terms. In many contexts, this is acceptable if the answer is given. But wait, look at the signs in the stated answer again. 1280x^4 (pos) - 256x^3 (neg) + 48x^2 (pos) - 8x (neg) + 1 (pos). The expansion of (1+u)^-2 is 1 - 2u + 3u^2 - 4u^3 + 5u^4. With u=4x: 1 - 8x + 3(16)x^2 - 4(64)x^3 + 5(256)x^4 = 1 - 8x + 48x^2 - 256x^3 + 1280x^4. The stated answer matches. The solution is correct. Why did I think it might fail? Perhaps because it doesn't explicitly list the terms in the text? But the 'Stated answer' is separate. The solution lines support the stated answer. It seems sound. Let me double check the 'adversarial' instruction. 'Find what is wrong'. Is there a subtle issue? Line 1: 'r = -2 and u = 4*x'. Correct. Line 2: Checks x^2 coeff. Correct. Line 3: Checks limit. Correct. The solution is a valid proof that the stated answer is correct. I will mark it pass.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the terms involving x, x^3, and x^4. Since u = 4x and r = -2, all terms in the binomial expansion (1+4x)^{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 the wrong signs for the terms (it lists alternating signs starting with positive for x^4, but the expansion of (1+4x)^{-2} hasgpt-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.