Calculate the Taylor’s inequality error bound M|x − a|^(n+1)/(n+1)! for a Taylor polynomial, or find the smallest degree n that meets a target error.

Use 0 for a Maclaurin series.

Largest |f⁽ⁿ⁺¹⁾(t)| between a and x. For degree search it must bound every derivative order.

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Taylor Inequality Error Formula

The following formula is used to calculate the Taylor inequality error.

|Rₙ(x)| ≤ E = (M × |x - a|ⁿ⁺¹) / ((n + 1)!)

Variables:

  • E is an upper bound on the absolute Taylor remainder, not the actual error
  • M bounds |f^(n+1)(t)| throughout the segment between a and x
  • r = |x − a| is the distance from the Taylor center to the evaluation point
  • (n+1)! is the factorial of n+1

To calculate the Taylor inequality error, multiply the maximum value of the (n+1)th derivative by the radius raised to the power of n+1, and then divide by the factorial of n+1.

What is Taylor Inequality Error?

Taylor inequality error is an estimation of the error made when using a Taylor polynomial to approximate a function. It is based on the remainder term of Taylor's theorem, which provides a bound on the error. This error bound is useful for determining the accuracy of the approximation and for deciding how many terms of the Taylor series are needed to achieve a desired level of precision.

How to Calculate Taylor Inequality Error?

The following steps outline how to calculate the Taylor Inequality Error.

  1. First, determine a valid upper bound M on the absolute (n+1)th derivative throughout the segment from a to x. In custom degree search the same M must cover every tested derivative order.
  2. Next, determine the distance r = |x − a|.
  3. Next, gather the formula from above: E = M * r^(n+1) / (n+1)!.
  4. Finally, calculate the Taylor Inequality Error (E).
  5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Upper bound M on the absolute (n+1)th derivative throughout the segment = 24

Radius (r) = 0.5

For n = 3, E = 24 × 0.5^4 / 4! = 0.0625, so |R₃(x)| ≤ 0.0625. This is a bound, not a claim that the actual error equals 0.0625.