Taylor series/R/One variable/Introduction/Section

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Let denote an interval,

an infinitely often differentiable function, and . Then

is called the Taylor series of in the point .

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.

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.

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 .