The binomial series
Problem 7.665 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt{4 x + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = 4*x.
- \[ \frac{4^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = -2 \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{10 x^{4} - 4 x^{3} + 2 x^{2} - 2 x + \sqrt{4 x + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - 10 x^{4} + 4 x^{3} - 2 x^{2} + 2 x + 1 + \cdots \)
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 lists terms in descending order of powers, but the coefficients are incorrect for the standard binomial expansion. Specifically, the coefficient for x^4 in the expansion of (1+4x)^(1/2) is 10, not -10. The solution's check (line 3) verifies that the polynomial 10x^4 - 4x^3 + 2x^2 - 2x + 1 approximates the function, but the stated answer has -10x^4. The sign error in the stated answer makes it incorrect.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The stated answer lists terms in descending order of powers, but the coefficients are incorrect for the standard binomial expansion. Specifically, the coefficient for x^4 in the expansion of (1+4x)^(1/2) is 10, not -10. The solution's check (line 3) verifies that the polynomial 10x^4 - 4x^3 + 2x^2 - 2x + 1 approximates the function, but the stated answer has -10x^4. The sign error in the stated answer makes it incorrect.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The stated answer has incorrect signs for the x^3 and x^4 terms (it lists -4x^3 and -10x^4, but the correct coefficients are positive). The solution's verification step (line 3) checks against a polynomial with the correct signs, contradicting the stated answer.gpt-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 lists terms in descending order of power, but the constant term is 1, not -10x^4. The series expansion of sqrt(4x+1) is 1 +gpt-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.