The binomial series
Problem 7.662 · medium
Find the first five terms of the binomial series for \( \displaystyle \sqrt{x + 1} \).
- (1 + u)ʳ = 1 + ru + r(r − 1)/2! u² + r(r − 1)(r − 2)/3! u³ + ⋯ with r = 1/2 and u = x.
- \[ \frac{1^{2} \left(-1 + 1 \cdot \frac{1}{2}\right)}{2 \cdot 2} = - \frac{1}{8} \]The x² coefficient, r(r − 1)/2 · c².✓ Proved
- \[ \lim_{x \to 0}\left(\frac{\frac{5 x^{4}}{128} - \frac{x^{3}}{16} + \frac{x^{2}}{8} - \frac{x}{2} + \sqrt{x + 1} - 1}{x^{4}}\right) = 0 \]These terms agree with the function to order x⁴.✓ Proved
Answer \( - \frac{5 x^{4}}{128} + \frac{x^{3}}{16} - \frac{x^{2}}{8} + \frac{x}{2} + 1 + \cdots \)
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 (error) — The limit in step 3 uses the wrong signs for the series terms; it does not correspond to the given binomial expansion and therefore does not verify the coefficients correctly.qwen3.6:27b-mlx: pass — The solution correctly identifies the binomial series formula and parameters. The algebraic checks confirm the coefficients and the order of approximation match the stated answer.
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 checks confirm the coefficients and the order of approximation match the stated answer.gpt-oss:20b: fail (error) 2026-10-18 — The limit in step 3 uses the wrong signs for the series terms; it does not correspond to the given binomial expansion and therefore does not verify the coefficients correctly.qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The stated answer has incorrect signs for the $x^2$ and $x^4$ terms. The binomial expansion of $(1+x)^{1/2}$ is $1 + x/2 - x^2/8 + x^3/16 - 5x^4/128gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the binomial series formula and parameters. The unchecked equations verify the coefficient calculation and the asymptotic agreement of the series with the function, confirming the result is correct.gpt-oss:20b: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution’s final check uses the wrong sign for the \(x^4\) term (it writes \(+5x^4/128\) instead of the correct \(-5x^4/128\)). Consequently the limit
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.