The Lagrange error bound
Problem 7.473 · 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 \( \displaystyle x = 1 \)?
- |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.
- \[ 1 \cdot \frac{1}{120960} = \frac{1}{120960} \]n = 8: the bound is 1/120960 < 1/100000.✓ Proved
- \[ 1 \cdot \frac{1}{13440} = \frac{1}{13440} \]n = 7: the bound is 1/13440, not below 1/100000.✓ Proved
Answer \( n = 8 \)
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: inconclusive — reviewer returned a non-object
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution uses M=3, which is a valid upper bound, but the arithmetic in line 2 is incorrect: 3/(9!) = 3/362880 = 1/120960, which is approximately 8.27e-6, not less than 1/100000 (1e-5). Thus, n=8 does not guarantee the error is less than 1/100000 using this bound; n=9 is required (3/10! = 3/3628800 = 1/1209600 < 1e-5). The conclusion n=8 is wrong.
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 2026-10-10 with SymPy 1.14.0.