The binomial series
Problem 7.666 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt{1 - x} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = -x.Reviewed
- \[ \frac{\left(-1\right)^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{1}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{128} + \frac{x^{3}}{16} + \frac{x^{2}}{8} + \frac{x}{2} + \sqrt{1 - x} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{5 x^{4}}{128} - \frac{x^{3}}{16} - \frac{x^{2}}{8} - \frac{x}{2} + 1 + \cdots \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic check for the x^2 coefficient is correct, and the limit check confirms the series expansion matches the function up to the required order.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the binomial series formula and parameters. The algebraic check for the x^2 coefficient is correct, and the limit check confirms the series expansion matches the function up to the required order.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial series formula and parameters. The provided equations verify the coefficient of the x² term and the overall accuracy of the first five terms up to order x⁴, which is sufficient to confirm the stated answer.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-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 states the binomial theorem and checks a coefficient, but it never actually derives or lists the first five terms of the series. The final sentence merely verifies the result without showing how the terms were obtained, leaving the answer incomplete and potentially confusing for a student.
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.