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Taylor Polynomial Error Bound

∣Rn(x)∣≤M(n+1)!∣x−a∣n+1|R_n(x)| \leq \frac{M}{(n+1)!}|x - a|^{n+1}

The Lagrange error bound (Taylor remainder): the error in the nth-degree Taylor approximation is bounded by M|x-a|^(n+1)/(n+1)!, where M is the max of |f⁽ⁿ⁺¹⁾| on the interval.

  • 2 worked examples
  • 18 in Sequences & Series

Worked examples

Bound the error in approximating e^0.1 using the 3rd-degree Maclaurin polynomial.
  1. f(x) = eˣ. f⁽⁴⁾(x) = eˣ. On [0, 0.1], max |f⁽⁴⁾| ≤ e^0.1 < 1.2
  2. |R₃| ≤ 1.2 · (0.1)⁴/4! = 1.2 · 0.0001/24 = 0.000005

Answer: |R₃| ≤ 5 × 10⁻⁶

1 more worked examplePremium

Bound the error in approximating sin(0.5) with the 3rd-degree Maclaurin polynomial.

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Common questions

What is the Taylor Polynomial Error Bound?

The Lagrange error bound (Taylor remainder): the error in the nth-degree Taylor approximation is bounded by M|x-a|^(n+1)/(n+1)!, where M is the max of |f⁽ⁿ⁺¹⁾| on the interval. It is one of the sequences & series formulas in the CalcRef reference.

How do you use the Taylor Polynomial Error Bound?

Worked example. Bound the error in approximating e^0.1 using the 3rd-degree Maclaurin polynomial. f(x) = eˣ. f⁽⁴⁾(x) = eˣ. On [0, 0.1], max |f⁽⁴⁾| ≤ e^0.1 < 1.2. |R₃| ≤ 1.2 · (0.1)⁴/4! = 1.2 · 0.0001/24 = 0.000005. Answer: |R₃| ≤ 5 × 10⁻⁶.

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