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Math Help - Taylor Series, Complex

  1. #1
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    Taylor Series, Complex

    Obtain the Taylor series e^z=e \cdot \sum_{n=0}^{\infty} \frac{(z-1)^n}{n!} (  |z - 1 | < \infty ) for the function f(z)=e^z by using f^{(n)}(1) where (n=1, 2, \ldots).

    I do not see how to do this. I was able to prove that by writing e^z=e^{z-1}e but I don't see how to do it that way. Thanks in advance for help.
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  2. #2
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    Quote Originally Posted by eskimo343 View Post
    Obtain the Taylor series e^z=e \cdot \sum_{n=0}^{\infty} \frac{(z-1)^n}{n!} (  |z - 1 | < \infty ) for the function f(z)=e^z by using f^{(n)}(1) where (n=1, 2, \ldots).

    I do not see how to do this. I was able to prove that by writing e^z=e^{z-1}e but I don't see how to do it that way. Thanks in advance for help.

    Well, since f^{(n)}(z)=f(z)\,\,\forall\,z\in \mathbb{C} , we get that f^{(n)}(1)=e\,\,\forall n\in \mathbb{N}, so...
    Tonio
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