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MATH
[1251]

Methods from Taylor's theorem

All methods from Taylor's theorem.

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Bound a Taylor approximation error

To prove that Pn(x)P_n(x) approximates f(x)f(x) within a tolerance ε\varepsilon:

  1. identify f(n+1)f^{(n+1)};
  2. find a bound MM for ∣f(n+1)(t)∣|f^{(n+1)}(t)| between aa and the relevant values of xx;
  3. use ∣f(x)−Pn(x)∣≤M∣x−a∣n+1(n+1)!;|f(x)-P_n(x)| \le \frac{M|x-a|^{n+1}}{(n+1)!};
  4. choose nn or the interval so that the right-hand side is at most ε\varepsilon.

The estimate is uniform on an interval when one MM and one distance bound work throughout that interval. It bounds the error from above; it is not the actual error.

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Bound a Taylor approximation error