Taylor series/R/One variable/Introduction/Section
Definition
Let denote an interval,
an infinitely often differentiable function, and . Then
Theorem
Let denote a power series which converges on the interval , and let
denote the function defined via fact. Then is infinitely often differentiable, and the Taylor series of in coincides with the given power series.
Proof
That is infinitely often differentiable, follows directly from fact by induction. Therefore, the Taylor series exists in particular in the point . Hence, we only have to show that the -th derivative has as its value. But this follows also from fact.
Example
We consider the function
given by
We claim that this function is infinitely often differentiable, which is only in not directly clear. We first show, by induction, that all derivatives of have the form with certain polynomials , and that therefore the limit for equals (see exercise and exercise). Therefore, the limit exists for all derivatives and is . So all derivatives in have value , and therefore the Taylor series in is just the zero series. However, the Function is in no neighborhood of the zero function, since .