Home›Calculus 2›The Lagrange error bound›Problem 7.475

The Lagrange error bound

Problem 7.475 · 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 \( \displaystyle x = \frac{1}{5} \)?

|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.
\[ 1 \cdot \frac{1}{125000} = \frac{1}{125000} \]
n = 4: the bound is 1/125000 < 1/100000.✓ Proved
\[ 1 \cdot \frac{1}{5000} = \frac{1}{5000} \]
n = 3: the bound is 1/5000, 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: pass
qwen3.6:27b-mlx: fail (error) — The solution uses M=3, which is a valid upper bound for e^x on [0, 1/5], but the calculation for n=4 is incorrect. With M=3, x=1/5, and n=4, the bound is 3*(1/5)^5 / 120 = 3/156250 ≈ 0.0000192, which is indeed less than 0.00001. However, the solution claims the bound is 1/125000 (which equals 0.000008). 1/125000 corresponds to using M=1 (since 1*(1/5)^5/120 = 1/156250? No, 1/125000 is 8/1000000. Let's recompute: 3/156250 = 0.0000192. 1/125000 = 0.000008. The value 1/125000 is not derived from M=3. It seems the solver used M=1 or made an arithmetic error. Actually, if M=e, e/156250 ≈ 0.000017. If M=3, 3/156250 ≈ 0.0000192. Both are < 0.00001. The stated value 1/125000 is mathematically inconsistent with M=3. Specifically, 3/156250 != 1/125000. 1/125000 = 1.25e-5. 3/156250 = 1.92e-5. The sentence claims the bound IS 1/125000, which is false given M=3. This is a calculation error in the explanation.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution uses M=3, which is a valid upper bound for e^x on [0, 1/5], but the calculation for n=4 is incorrect. With M=3, x=1/5, and n=4, the bound is 3*(1/5)^5 / 120 = 3/156250 ≈ 0.0000192, which is indeed less than 0.00001. However, the solution claims the bound is 1/125000 (which equals 0.000008). 1/125000 corresponds to using M=1 (since 1*(1/5)^5/120 = 1/156250? No, 1/125000 is 8/1000000. Let's recompute: 3/156250 = 0.0000192. 1/125000 = 0.000008. The value 1/125000 is not derived from M=3. It seems the solver used M=1 or made an arithmetic error. Actually, if M=e, e/156250 ≈ 0.000017. If M=3, 3/156250 ≈ 0.0000192. Both are < 0.00001. The stated value 1/125000 is mathematically inconsistent with M=3. Specifically, 3/156250 != 1/125000. 1/125000 = 1.25e-5. 3/156250 = 1.92e-5. The sentence claims the bound IS 1/125000, which is false given M=3. This is a calculation error in the explanation.
qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution uses M=3, which is a valid upper bound for e^x on [0, 1/5], but the resulting error bound for n=4 is 3*(1/5)^5/120 = 3/93750 = 1/31250, not 1/125000. The calculation in line 2 is arithmetically incorrect (it seems to have used M=1 or ignored the factor of 3 and factorial incorrectly), leading to a false justification for n=4. Although n=4 is indeed the correct answer (since n=3 gives 3*(1/5)^4/24 = 3/7500 = 1/2500 > 1/100000), the derivation provided is mathematically flawed.
gpt-oss:20b: inconclusive 2026-10-10 — 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 2026-10-10 with SymPy 1.14.0.