The Lagrange error bound
Problem 7.268 · easy
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle \cos{\left(x \right)} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{100000} \) at \( \displaystyle x = 1 \)?
- |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
- \[ 1 \cdot \frac{1}{362880} = \frac{1}{362880} \]n = 8: the bound is 1/362880 < 1/100000.✓ Proved
- \[ 1 \cdot \frac{1}{40320} = \frac{1}{40320} \]n = 7: the bound is 1/40320, not below 1/100000.✓ Proved
Answer \( n = 8 \)
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 for the Maclaurin series of cos(x). It correctly identifies M=1 and verifies that n=8 satisfies the error condition while n=7 does not.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Lagrange error bound formula for the Maclaurin series of cos(x). It correctly identifies M=1 and verifies that n=8 satisfies the error condition while n=7 does not.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly applies the Lagrange error bound formula, using n! instead of (n+1)! in the denominator for the checked values (e.g., 8! = 40320, not 362880). This arithmetic error leads to the wrong conclusion that n=8 is sufficient, when in fact n=9 is required to guarantee the error is less than 1/100000.gpt-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.