The binomial series
Problem 7.634 · medium
Find the first five terms of the binomial series for \( \displaystyle \left(3 x + 1\right)^{\frac{2}{3}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 2/3 and u = 3*x.
- \[ \frac{2 \cdot 3^{2} \left(-1 + \frac{2}{3}\right)}{2 \cdot 3} = -1 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{7 x^{4}}{3} - \frac{4 x^{3}}{3} + x^{2} - 2 x + \left(3 x + 1\right)^{\frac{2}{3}} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{7 x^{4}}{3} + \frac{4 x^{3}}{3} - x^{2} + 2 x + 1 + \cdots \)
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 $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{2/3}$ yields positive coefficients for all terms because the factors in the numerator of the binomial coefficients alternate in sign but are divided by negative denominators or result in positive products when combined with the positive powers of $3x$. Specifically, the $x^3$ term is $+4/3 x^3$ and the $x^4$ term is $+7/3 x^4$, whereas the stated answer lists them as negative.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer has incorrect signs for the $x^3$ and $x^4$ terms. The binomial expansion of $(1+3x)^{2/3}$ yields positive coefficients for all terms because the factors in the numerator of the binomial coefficients alternate in sign but are divided by negative denominators or result in positive products when combined with the positive powers of $3x$. Specifically, the $x^3$ term is $+4/3 x^3$ and the $x^4$ term is $+7/3 x^4$, whereas the stated answer lists them as negative.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial expansion formula and the substitution u = 3x. The algebraic checks confirm the coefficients match the stated answer.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to explicitly calculate and list the first five terms as requested, instead relying on an unchecked limit check that does not demonstrate the derivation of the coefficients. Additionally, the stated answer lists terms in descending order of power, which is unconventional for a series expansion around x=0, and the solution text does not address this ordering or the calculation of the x^3 and x^4 coefficients.gpt-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 2026-10-18 with SymPy 1.14.0.