The Lagrange error bound
Problem 7.259 · easy
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \sin{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000} \) at \( \displaystyle x = \frac{1}{2} \)?
- |Rₙ(x)| ≤ M|x|ⁿ⁺¹/(n + 1)!, where M bounds |f⁽ⁿ⁺¹⁾|; here M = 1 because |sin| and |cos| are at most 1.Reviewed
- \[ 1 \cdot \frac{1}{3840} = \frac{1}{3840} \]n = 4: the bound is 1/3840 < 1/1000.✓ Proved
- \[ 1 \cdot \frac{1}{384} = \frac{1}{384} \]n = 3: the bound is 1/384, not below 1/1000.✓ Proved
Answer \( n = 4 \)
Lines: 2 proved, 1 reviewed. 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.
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 | the bound recomputed at n and n − 1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Lagrange error bound formula with M=1. It correctly identifies that n=4 satisfies the condition (1/3840 < 1/1000) while n=3 does not (1/384 > 1/1000), making n=4 the minimal degree.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Lagrange error bound formula with M=1. It correctly identifies that n=4 satisfies the condition (1/3840 < 1/1000) while n=3 does not (1/384 > 1/1000), making n=4 the minimal degree.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly applies the Lagrange error bound formula, using |x|^(n+1)/(n+1)! instead of |x|^(n+1)/(n+1)! * M, but more critically, it fagpt-oss:20b: pass 2026-10-04
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/lagrange_error, checked 2026-10-04 with SymPy 1.14.0.