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I have learned that we can prove Euler's formula by using Taylor series, as shown on wiki: Euler's Formula.

I have a question. As wiki says: In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

Since Taylor series is only the expression of a function at a single point. And the first proof of Euler's Formula is using the Taylor Series at point 0. Why it can be used to prove Euler's Formula.

Dongguo
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The Taylor series (the formula) depends only from the values of the function's derivatives at a single point ($a$ is fixed). But the Taylor series (the function) is a... function and depends of a variable ($x$ varies).

Martín-Blas Pérez Pinilla
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  • Thanks for your help. I spent several days studying and now I almost understand it. But I still have a question, is there any sense in terms of geometry or physics. Because it is quite difficult for me to understand a function,which is periodic such as sin(x), can be expressed by a bunch of polynomials. – Dongguo Jan 22 '15 at 13:02
  • @Dongguo, any approximation of $\sin$ by a polynomial is good only in a finite interval. See the diagram in http://en.wikipedia.org/wiki/Taylor_series. – Martín-Blas Pérez Pinilla Jan 23 '15 at 07:09