The binomial series
Problem 7.668 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt[3]{x + 1}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/3 and u = x.
- \[ \frac{1^{2} \left(1 \cdot \frac{1}{3} + 1\right)}{2 \cdot 3} = \frac{2}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{243} + \frac{14 x^{3}}{81} - \frac{2 x^{2}}{9} + \frac{x}{3} - 1 + \frac{1}{\sqrt[3]{x + 1}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{243} - \frac{14 x^{3}}{81} + \frac{2 x^{2}}{9} - \frac{x}{3} + 1 + \cdots \)
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: fail (misleading) — The solution verifies the coefficients but never actually lists the first five terms of the series, so it does not answer the problem as stated.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial expansion parameters and verifies the coefficients against the target function. The stated answer matches the derived terms.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients against the target function. The stated answer matches the derived terms.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution verifies the coefficients but never actually lists the first five terms of the series, so it does not answer the problem as stated.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer has the wrong signs for the odd-powered terms (it lists -x/3 and -14x^3/81, but the expansion of (1+x)^(-1/3) has +x/3 and +14x^3/81). The solution's verification in line 3 checks a polynomial with flipped signs against the function, which would not yield a limit of 0, indicating the stated answer is incorrect and the verification is flawed.gpt-oss:20b: fail (style) 2026-10-16 — The solution states the binomial expansion and verifies the x² coefficient, but it never actually lists the first five terms of the series, leaving the answer incomplete. The check in step 3 confirms agreement up to x⁴, yet the missing explicit terms could mislead a student into thinking the work is finished.qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients match the stated answer.gpt-oss:20b: fail (style) 2026-10-15 — The solution correctly states the binomial expansion and verifies the x² coefficient, but it does not actually list the first five terms of the series as required by the problem. The check in line 3 only confirms agreement up to x⁴ without providing the missing coefficients, so the answer is incomplete rather than incorrect.
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.