The binomial series
Problem 7.664 · medium
Find the first five terms of the binomial series for \( \displaystyle \frac{1}{\sqrt{1 - 2 x}} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = -1/2 and u = -2*x.
- \[ \frac{\left(-2\right)^{2} \left(1 \cdot \frac{1}{2} + 1\right)}{2 \cdot 2} = \frac{3}{2} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- \frac{35 x^{4}}{8} - \frac{5 x^{3}}{2} - \frac{3 x^{2}}{2} - x - 1 + \frac{1}{\sqrt{1 - 2 x}}}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( \frac{35 x^{4}}{8} + \frac{5 x^{3}}{2} + \frac{3 x^{2}}{2} + x + 1 + \cdots \)
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 solution claims to find the first five terms but only explicitly calculates and verifies coefficients up to x^2 (three terms). It fails to compute or state the coefficients for x^3 and x^4, which are required for the first five terms.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution claims to find the first five terms but only explicitly calculates and verifies coefficients up to x^2 (three terms). It fails to compute or state the coefficients for x^3 and x^4, which are required for the first five terms.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the binomial expansion parameters and verifies the coefficients. The setup is sound and the conclusion follows.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution verifies a single coefficient and a limit test but never actually lists the first five terms of the series, so it does not answer the problem and could mislead a student into thinking the work is complete.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The setup incorrectly substitutes u = -2x instead of u = 2x. The function is (1 - 2x)^(-1/2), which matches the binomial form (1 + u)^r with u = -2xgpt-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.