Home›Calculus 2›The Lagrange error bound›Problem 7.263

The Lagrange error bound

Problem 7.263 · easy

What degree \( \displaystyle n \) of Maclaurin polynomial for \( \displaystyle e^{x} \) guarantees, by the Lagrange error bound, an error less than \( \displaystyle \frac{1}{1000000} \) at \( \displaystyle x = \frac{1}{5} \)?

|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.
\[ 1 \cdot \frac{1}{3750000} = \frac{1}{3750000} \]
n = 5: the bound is 1/3750000 < 1/1000000.✓ Proved
\[ 1 \cdot \frac{1}{125000} = \frac{1}{125000} \]
n = 4: the bound is 1/125000, not below 1/1000000.✓ Proved

Answer \( n = 5 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly concludes n=5 is the answer, but n=4 also satisfies the condition (1/125000 < 1/1000000). The question asks for the degree that guarantees the error bound, implying the smallest such n, which is 4.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution incorrectly concludes n=5 is the answer, but n=4 also satisfies the condition (1/125000 < 1/1000000). The question asks for the degree that guarantees the error bound, implying the smallest such n, which is 4.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer returned a non-object
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.