The Lagrange error bound
Problem 7.344 · easy
What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{100000} \) at \( \displaystyle x = \frac{1}{10} \)?
- |Rₙ(x)| ≤ M|x|ⁿ⁺¹/(n + 1)!, where M bounds |f⁽ⁿ⁺¹⁾|; here M = 3 because on [0, 1] every derivative of eˣ is at most e < 3.
- \[ 1 \cdot \frac{1}{4000000} = \frac{1}{4000000} \]n = 4: the bound is 1/4000000 < 1/100000.✓ Proved
- \[ 1 \cdot \frac{1}{80000} = \frac{1}{80000} \]n = 3: the bound is 1/80000, not below 1/100000.✓ Proved
Answer \( n = 4 \)
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.
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 | the bound recomputed at n and n − 1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution uses M=3, which is a valid upper bound for e^x on [0, 1/10], but the resulting error bound for n=3 is 3*(1/10)^4/24 = 1/8000, not 1/80000 as claimed. The arithmetic in the 'proved' equations is incorrect, leading to a false justification for why n=3 fails.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution uses M=3, which is a valid upper bound for e^x on [0, 1/10], but the resulting error bound for n=3 is 3*(1/10)^4/24 = 1/8000, not 1/80000 as claimed. The arithmetic in the 'proved' equations is incorrect, leading to a false justification for why n=3 fails.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer returned a non-object
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 2026-10-06 with SymPy 1.14.0.