Mathematics
July 22, 2026 · 9 min read
Lagrange Error Bound: Guaranteeing Taylor Polynomial Accuracy
Calculate the Lagrange error bound for Taylor polynomial approximations. Learn the formula, find maximum errors, and ensure approximation accuracy.
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When we approximate a function using a Taylor polynomial, we need to know how accurate that approximation is. The Lagrange error bound provides a rigorous upper limit on the maximum possible error, making it essential for numerical analysis and engineering applications.
The Formula
|Rₙ(x)| ≤ M/(n+1)! × |x - a|ⁿ⁺¹
Where M is the maximum value of |f⁽ⁿ⁺¹⁾(t)| on the interval between a and x.
Worked Example
Approximate e^0.1 using a 3rd degree Taylor polynomial centered at 0:
- f(x) = eˣ, so all derivatives are eˣ
- M = max|eᵗ| for t ∈ [0, 0.1] = e^0.1 ≈ 1.10517
- Error bound = 1.10517/4! × (0.1)⁴ ≈ 4.6 × 10⁻⁶
Key Insights
- The error bound decreases factorially with increasing degree
- Being closer to the center (smaller |x-a|) reduces error
- The bound is usually larger than the actual error