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The binomial series

Problem 7.629 · medium

Find the first five terms of the binomial series for \( \displaystyle \left(4 x + 1\right)^{\frac{3}{2}} \).

(1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 3/2 and u = 4*x.
\[ \frac{3 \cdot 4^{2} \left(-1 + \frac{3}{2}\right)}{2 \cdot 2} = 6 \]
The x² coefficient, r(r − 1)/2 · c².✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 6 x^{4} + 4 x^{3} - 6 x^{2} - 6 x + \left(4 x + 1\right)^{\frac{3}{2}} - 1}{x^{4}}\right) = 0 \]
These terms agree with the function to order x⁴.✓ Proved

Answer \( 6 x^{4} - 4 x^{3} + 6 x^{2} + 6 x + 1 + \cdots \)

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: fail (misleading) — The solution states that the terms agree with the function to order x⁴ but never actually lists the first five terms (x⁴, x³, x², x, 1). Without showing the coefficients of x⁴, x³, and x, the conclusion is unsupported and would mislead a student.
qwen3.6:27b-mlx: fail (error) — The stated answer has incorrect signs for the odd-power terms (x and x^3). The binomial expansion of (1+4x)^(3/2) yields positive coefficients for all terms because the ratio of consecutive coefficients involves (r-k)/(k+1) which remains positive for the first few terms, or simply by calculating: 1 + 6x + 6x^2 - 4x^3 + 6x^4. The provided solution claims the answer is 1 + 6x + 6x^2 - 4x^3 + 6x^4, but the check in line 3 verifies the polynomial -6x^4 + 4x^3 - 6x^2 - 6x + 1, which corresponds to 1 - 6x - 6x^2 + 4x^3 - 6x^4, contradicting the stated answer and the actual expansion.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The stated answer has incorrect signs for the odd-power terms (x and x^3). The binomial expansion of (1+4x)^(3/2) yields positive coefficients for all terms because the ratio of consecutive coefficients involves (r-k)/(k+1) which remains positive for the first few terms, or simply by calculating: 1 + 6x + 6x^2 - 4x^3 + 6x^4. The provided solution claims the answer is 1 + 6x + 6x^2 - 4x^3 + 6x^4, but the check in line 3 verifies the polynomial -6x^4 + 4x^3 - 6x^2 - 6x + 1, which corresponds to 1 - 6x - 6x^2 + 4x^3 - 6x^4, contradicting the stated answer and the actual expansion.
gpt-oss:20b: fail (misleading) 2026-10-17 — The solution states that the terms agree with the function to order x⁴ but never actually lists the first five terms (x⁴, x³, x², x, 1). Without showing the coefficients of x⁴, x³, and x, the conclusion is unsupported and would mislead a student.
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The stated answer has incorrect signs for the odd-power terms; the expansion of (1+4x)^(3/2) yields positive coefficients for all terms, but the answer lists negative coefficients for x and x^3. Additionally, the terms are listed in descending order of power, which is unconventional for a series expansion around 0.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The stated answer has incorrect signs for the odd-powered terms (x and x^3) and the x^4 term. The binomial expansion of (1+4x)^(3/2) yields positive coefficients for all terms since the factors in the numerator alternate in sign but are multiplied by positive powers of 4, resulting in all positive terms: 1 + 6x + 6x^2 + 4x^3 + ... The solution's verification step (line 3) also contains a sign error in the polynomial being subtracted, incorrectly suggesting the negative terms are correct.
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 2026-10-17 with SymPy 1.14.0.