Taylor series

Taylor's theorem

Taylor's theorem describes the margin of error for a Taylor series. It states that if is -differentiable, then for any there exists a such that #m/thm/anal

where is the Taylor series about to order and the Lagrange remainder is given by

Therefore a maximum error is given by the maximum of with respect to .

Proof1

Let

i.e. so that

We want to show there exists a such that . Now define

Then , so by Rolle's theorem there exists some such that . Now

so for implies , as required.

#state/tidy | #SemBr | #lang/en

Footnotes

  1. 1953. A Proof of Taylor's Formula ↩