Sequences & Series · Power & Taylor Series
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.
- 0 variables
- 2 worked examples
- 18 in Sequences & Series
Worked examples
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
1 more worked examplePremium
Bound the error in approximating sin(0.5) with the 3rd-degree Maclaurin polynomial.
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