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 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 .

Proof ¹

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.

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