The binomial series
Problem 7.663 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{4 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = 4*x.
- \[ \frac{4^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = \frac{32}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{8960 x^{4}}{243} + \frac{896 x^{3}}{81} - \frac{32 x^{2}}{9} + \frac{4 x}{3} - 1 + \frac{1}{\sqrt[3]{4 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{8960 x^{4}}{243} - \frac{896 x^{3}}{81} + \frac{32 x^{2}}{9} - \frac{4 x}{3} + 1 + \cdots \)
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 odd-power terms; the binomial expansion of (1+4x)^{-1/3} yields positive coefficients for all terms, but the answer lists -4x/3 and -896x^3/81. Additionally, the solution text claims the terms agree with the function, which is false given the sign errors.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer has incorrect signs for the odd-power terms; the binomial expansion of (1+4x)^{-1/3} yields positive coefficients for all terms, but the answer lists -4x/3 and -896x^3/81. Additionally, the solution text claims the terms agree with the function, which is false given the sign errors.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 the wrong signs for the odd-power terms (it lists -4x/3 instead of +4x/3, and -896x^3/81 instead of +896x^3/81). The binomialgpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The stated answer has incorrect signs for the odd-power terms. The binomial expansion of (1+u)^(-1/3) yields negative coefficients for all powers of u (since r<0 and r-1, r-2 etc are negative, but the product r(r-1)... alternates sign? No. r=-1/3. Term 1: 1. Term 2: (-1/3)u. Term 3: (-1/3)(-4/3)/2 u^2 = (4/9)/2 u^2 = 2/9 u^2. Term 4: (-1/3)(-4/3)(-7/3)/6 u^3 = (-28/27)/6 u^3 = -14/81 u^3. Term 5: (-1/3)(-4/3)(-7/3)(-10/3)/24 u^4 = (280/81)/24 u^4 = 35/243 u^4. Substituting u=4x: 1 - 4/3 x + 2/9(16)x^2 - 14/81(64)x^3 + 35/243(256)x^4 = 1 - 4/3 x + 32/9 x^2 - 896/81 x^3 + 8960/243 x^4. The stated answer has +4x/3 and -8960/243 x^4? No, it has +8960/243 x^4 but -896/81 x^3 and +4x/3. Wait, the stated answer is: 8960/243 x^4 - 896/81 x^3 + 32/9 x^2 - 4/3 x + 1. This matches my calculation: 1 - 4/3 x + 32/9 x^2 - 896/81 x^3 + 8960/243 x^4. The signs in the stated answer are correct. Why did I think they were wrong? Let me re-read the stated answer carefully. Stated: 8960/243 x^4 - 896/81 x^3 + 32/9 x^2 - 4/3 x + 1. My calc: 1 - 4/3 x + 32/9 x^2 - 896/81 x^3 + 8960/243 x^4. They are identical. So the answer is correct. The solution text says 'r = -1/3 and u = 4*x'. This is correct. The equations check out. Is there a subtle error? The problem asks for the first five terms. The solution provides them. The setup is correct. The check is correct. Wait, look at the first sentence again. '(1 + u)ʳ = ... with r = -1/3 and u = 4*x'. This is correct. Is there any issue with the domain or convergence? The problem just asks for the series terms. The solution is sound. Why would it fail? Let me double check the coefficients. r = -1/3. c1 = -1/3. c2 = (-1/3)(-4/3)/2 = 2/9. c3 = (-1/3)(-4/3)(-7/3)/6 = -14/81. c4 = (-1/3)(-4/3)(-7/3)(-10/3)/24 = 35/243. u = 4x. u^2 = 16x^2. u^3 = 64x^3. u^4 = 256x^4. Term 0: 1. Term 1: -1/3 * 4x = -4/3 x. Term 2: 2/9 * 16x^2 = 32/9 x^2. Term 3: -14/81 * 64x^3 = -896/81 x^3. Term 4: 35/243 * 256x^4 = 8960/243 x^4. The stated answer is: 8960/243 x^4 - 896/81 x^3 + 32/9 x^2 - 4/3 x + 1. This is exactly correct. The solution is correct. I will mark it as pass.gpt-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.