The binomial series
Problem 7.719 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt[3]{2 x + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/3 and u = 2*x.
- \[ \frac{2^{2} \left(-1 + 1 \cdot \frac{1}{3}\right)}{2 \cdot 3} = - \frac{4}{9} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{160 x^{4}}{243} - \frac{40 x^{3}}{81} + \frac{4 x^{2}}{9} - \frac{2 x}{3} + \sqrt[3]{2 x + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{160 x^{4}}{243} + \frac{40 x^{3}}{81} - \frac{4 x^{2}}{9} + \frac{2 x}{3} + 1 + \cdots \)
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 solution claims to find the first five terms but only explicitly calculates or verifies coefficients for terms up to x^2 (and implicitly x^4 via limit). It fails to derive or state the coefficients for x^3 and x^4, which are required for the 'first five terms' (1, x, x^2, x^3, x^4). The limit check confirms the final answer is correct, but the solution text does not show the work for the missing terms, making it an incomplete derivation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution claims to find the first five terms but only explicitly calculates or verifies coefficients for terms up to x^2 (and implicitly x^4 via limit). It fails to derive or state the coefficients for x^3 and x^4, which are required for the 'first five terms' (1, x, x^2, x^3, x^4). The limit check confirms the final answer is correct, but the solution text does not show the work for the missing terms, making it an incomplete derivation.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial expansion parameters and verifies the resulting polynomial terms against the function. Although the intermediate steps for coefficients are omitted, the final verification confirms the correctness of the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. The stated answer matches the derived terms.gpt-oss:20b: pass 2026-10-16
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 2026-10-19 with SymPy 1.14.0.