The binomial series
Problem 7.660 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{3 x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = 3*x.
- \[ \frac{3^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = 2 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{3} + \frac{14 x^{3}}{3} - 2 x^{2} + x - 1 + \frac{1}{\sqrt[3]{3 x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{3} - \frac{14 x^{3}}{3} + 2 x^{2} - x + 1 + \cdots \)
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 all terms except the constant term. The binomial expansion of (1+3x)^(-1/3) yields alternating signs starting with positive, i.e., 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4, whereas the stated answer lists them as 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4 but written in reverse order with flipped signs for the higher order terms in the provided text (specifically, the stated answer shows +35/3 x^4 - 14/3 x^3 + 2x^2 - x + 1, which is actually correct if read as a polynomial, but wait: the standard expansion is 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. The stated answer is 35/3 x^4 - 14/3 x^3 + 2x^2 - x + 1. This is the same polynomial. Let me re-read carefully. Expansion: Term 0: 1 Term 1: (-1/3)(3x) = -x Term 2: (-1/3)(-4/3)/2 * (3x)^2 = (4/9)/2 * 9x^2 = 2x^2 Term 3: (-1/3)(-4/3)(-5/3)/6 * (3x)^3 = (-20/27)/6 * 27x^3 = -20/6 x^3 = -10/3 x^3? No. Let's recompute Term 3 coefficient: r(r-1)(r-2)/3! = (-1/3)(-4/3)(-5/3)/6 = -20/27 / 6 = -20/162 = -10/81. Multiply by u^3 = (3x)^3 = 27x^3. Coeff = -10/81 * 27 = -10/3. Wait, the stated answer has -14/3 x^3. Let's check Term 4. r(r-1)(r-2)(r-3)/4! = (-1/3)(-4/3)(-5/3)(-6/3)/24 = (120/81)/24 = 120/1944 = 5/81. Multiply by u^4 = (3x)^4 = 81x^4. Coeff = 5/81 * 81 = 5. So the series is 1 - x + 2x^2 - 10/3 x^3 + 5 x^4. The stated answer is 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. These are different. The stated answer is incorrect. The solution claims the terms agree to order x^4 via a limit check, but the limit check equation in line 3 uses the coefficients from the *stated answer* (35/3, 14/3, etc) and claims the limit is 0. If the limit is 0, then the stated answer IS correct. But my manual calculation says it's not. Let me re-calculate carefully. Function: (1+3x)^(-1/3). Binomial series: sum binom(r, k) (3x)^k. k=0: 1. k=1: (-1/3) * 3x = -x. k=2: (-1/3)(-4/3)/2 * 9x^2 = (4/18)*9 x^2 = 2x^2. k=3: (-1/3)(-4/3)(-5/3)/6 * 27x^3 = (-20/27)/6 * 27 x^3 = -20/6 x^3 = -10/3 x^3. k=4: (-1/3)(-4/3)(-5/3)(-6/3)/24 * 81x^4 = (120/81)/24 * 81 x^4 = 120/24 x^4 = 5 x^4. Correct series: 1 - x + 2x^2 - 10/3 x^3 + 5 x^4. Stated answer: 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. The stated answer is wrong. The solution's line 3 claims a limit is 0, implying the stated answer is correct. This is a contradiction. The limit of (Stated - Actual)/x^4 would NOT be 0. Therefore, the claim in line 3 is false, or the 'proved' tag is lying/misleading about the result of the CAS check. If the CAS checked the limit of (Stated - Function)/x^4, it would return a non-zero value (specifically, (5 - 35/3) = -20/3 for the x^4 term difference? No, the limit is the coefficient of x^4 in the difference. Diff = (5 - 35/3)x^4 + ... = -20/3 x^4. Limit is -20/3 != 0. Thus, the solution asserts a falsehood in line 3 (that the limit is 0) to justify an incorrect stated answer. This is an error.
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+3x)^(-1/3) yields alternating signs starting with positive, i.e., 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4, whereas the stated answer lists them as 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4 but written in reverse order with flipped signs for the higher order terms in the provided text (specifically, the stated answer shows +35/3 x^4 - 14/3 x^3 + 2x^2 - x + 1, which is actually correct if read as a polynomial, but wait: the standard expansion is 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. The stated answer is 35/3 x^4 - 14/3 x^3 + 2x^2 - x + 1. This is the same polynomial. Let me re-read carefully. Expansion: Term 0: 1 Term 1: (-1/3)(3x) = -x Term 2: (-1/3)(-4/3)/2 * (3x)^2 = (4/9)/2 * 9x^2 = 2x^2 Term 3: (-1/3)(-4/3)(-5/3)/6 * (3x)^3 = (-20/27)/6 * 27x^3 = -20/6 x^3 = -10/3 x^3? No. Let's recompute Term 3 coefficient: r(r-1)(r-2)/3! = (-1/3)(-4/3)(-5/3)/6 = -20/27 / 6 = -20/162 = -10/81. Multiply by u^3 = (3x)^3 = 27x^3. Coeff = -10/81 * 27 = -10/3. Wait, the stated answer has -14/3 x^3. Let's check Term 4. r(r-1)(r-2)(r-3)/4! = (-1/3)(-4/3)(-5/3)(-6/3)/24 = (120/81)/24 = 120/1944 = 5/81. Multiply by u^4 = (3x)^4 = 81x^4. Coeff = 5/81 * 81 = 5. So the series is 1 - x + 2x^2 - 10/3 x^3 + 5 x^4. The stated answer is 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. These are different. The stated answer is incorrect. The solution claims the terms agree to order x^4 via a limit check, but the limit check equation in line 3 uses the coefficients from the *stated answer* (35/3, 14/3, etc) and claims the limit is 0. If the limit is 0, then the stated answer IS correct. But my manual calculation says it's not. Let me re-calculate carefully. Function: (1+3x)^(-1/3). Binomial series: sum binom(r, k) (3x)^k. k=0: 1. k=1: (-1/3) * 3x = -x. k=2: (-1/3)(-4/3)/2 * 9x^2 = (4/18)*9 x^2 = 2x^2. k=3: (-1/3)(-4/3)(-5/3)/6 * 27x^3 = (-20/27)/6 * 27 x^3 = -20/6 x^3 = -10/3 x^3. k=4: (-1/3)(-4/3)(-5/3)(-6/3)/24 * 81x^4 = (120/81)/24 * 81 x^4 = 120/24 x^4 = 5 x^4. Correct series: 1 - x + 2x^2 - 10/3 x^3 + 5 x^4. Stated answer: 1 - x + 2x^2 - 14/3 x^3 + 35/3 x^4. The stated answer is wrong. The solution's line 3 claims a limit is 0, implying the stated answer is correct. This is a contradiction. The limit of (Stated - Actual)/x^4 would NOT be 0. Therefore, the claim in line 3 is false, or the 'proved' tag is lying/misleading about the result of the CAS check. If the CAS checked the limit of (Stated - Function)/x^4, it would return a non-zero value (specifically, (5 - 35/3) = -20/3 for the x^4 term difference? No, the limit is the coefficient of x^4 in the difference. Diff = (5 - 35/3)x^4 + ... = -20/3 x^4. Limit is -20/3 != 0. Thus, the solution asserts a falsehood in line 3 (that the limit is 0) to justify an incorrect stated answer. This is an error.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 all terms except the constant term. The binomial expansion of (1+3x)^(-1/3) yields alternating signs startgpt-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 incorrect signs for all terms except the constant term. The binomial expansion of (1+3x)^(-1/3) yields alternating signs startgpt-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.