The Lagrange error bound
Problem 7.520 · 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 \( \displaystyle x = \frac{1}{5} \)?
- |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
- \[ 1 \cdot \frac{1}{15000} = \frac{1}{15000} \]n = 3: the bound is 1/15000 < 1/1000.✓ Proved
- \[ 1 \cdot \frac{1}{750} = \frac{1}{750} \]n = 2: the bound is 1/750, not below 1/1000.✓ Proved
Answer \( n = 3 \)
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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution incorrectly applies the Lagrange error bound formula, using n+1 in the exponent and factorial instead of k (the degree of the first non-zero term after n). For n=2, the error bound is actually |x|^3/3! = 1/3750, which is less than 1/1000, so n=2 is sufficient. The solution's calculation for n=2 yields 1/750, which is incorrect for the standard Lagrange bound (it seems to have used n+1=3 in the denominator but n=2 in the numerator power inconsistently or made an arithmetic error).
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 2026-10-12 with SymPy 1.14.0.