The binomial series
Problem 7.589 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\left(x + 1\right)^{2}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -2 and u = x.
- \[ - \frac{2 \cdot 1^{2} \left(-2 - 1\right)}{2} = 3 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 5 x^{4} + 4 x^{3} - 3 x^{2} + 2 x - 1 + \frac{1}{\left(x + 1\right)^{2}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( 5 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 + \cdots \)
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 the terms (e.g., +5x^4 instead of -5x^4). The binomial series for (1+x)^-2 is an alternating series 1 - 2x + 3x^2 - 4x^3 + 5x^4..., but the provided answer lists them with reversed signs for the odd powers and reversed signs for the even powers relative to the standard expansion, or simply has the wrong signs entirely. Specifically, the coefficient of x is -2, not -2x in the answer? Wait, the answer is 5x^4 - 4x^3 + 3x^2 - 2x + 1. Let's check: (1+x)^-2 = 1 - 2x + 3x^2 - 4x^3 + 5x^4. The stated answer is 1 - 2x + 3x^2 - 4x^3 + 5x^4. This matches. Wait, let me re-read carefully. Stated answer: 5 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1. This is exactly 1 - 2x + 3x^2 - 4x^3 + 5x^4. So the answer is correct. Why did I think it was wrong? Ah, I misread the signs in my head. 1 - 2x + 3x^2 - 4x^3 + 5x^4. The stated answer is 5x^4 - 4x^3 + 3x^2 - 2x + 1. This is the same polynomial. So the answer is correct. The solution steps are minimal but correct. Step 1 sets up the formula. Step 2 checks a coefficient. Step 3 checks the limit of the difference divided by x^4, which confirms the Taylor polynomial up to order 4. The limit being 0 means the error is o(x^4), so the terms up to x^4 are correct. The solution is sound.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer has incorrect signs for the terms (e.g., +5x^4 instead of -5x^4). The binomial series for (1+x)^-2 is an alternating series 1 - 2x + 3x^2 - 4x^3 + 5x^4..., but the provided answer lists them with reversed signs for the odd powers and reversed signs for the even powers relative to the standard expansion, or simply has the wrong signs entirely. Specifically, the coefficient of x is -2, not -2x in the answer? Wait, the answer is 5x^4 - 4x^3 + 3x^2 - 2x + 1. Let's check: (1+x)^-2 = 1 - 2x + 3x^2 - 4x^3 + 5x^4. The stated answer is 1 - 2x + 3x^2 - 4x^3 + 5x^4. This matches. Wait, let me re-read carefully. Stated answer: 5 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1. This is exactly 1 - 2x + 3x^2 - 4x^3 + 5x^4. So the answer is correct. Why did I think it was wrong? Ah, I misread the signs in my head. 1 - 2x + 3x^2 - 4x^3 + 5x^4. The stated answer is 5x^4 - 4x^3 + 3x^2 - 2x + 1. This is the same polynomial. So the answer is correct. The solution steps are minimal but correct. Step 1 sets up the formula. Step 2 checks a coefficient. Step 3 checks the limit of the difference divided by x^4, which confirms the Taylor polynomial up to order 4. The limit being 0 means the error is o(x^4), so the terms up to x^4 are correct. The solution is sound.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer has incorrect signs for the odd powers of x. The binomial expansion of (1+x)^{-2} is 1 - 2x + 3x^2 - 4x^3 + 5x^4, but the stated answer lists -2x - 4x^3. Additionally, the solution text claims the terms agree to order x^4, but the provided equation checks a polynomial with signs opposite to the standard expansion (or matching the stated answer's errors), failing to verify the correct coefficients.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the odd powers of x. The binomial expansion of (1+x)^{-2} is 1 - 2x + 3x^2 - 4x^3 + 5x^4 - ..., but the stgpt-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.