Results 1 to 2 of 2

Math Help - Exponential Taylor Series

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    88

    Exponential Taylor Series

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

    So it would seem to me that f^{(n)}(1)=e, \forall n, but clearly that can't be the case. Clearly that's the correct Taylor series, since it's just e times the Taylor series for e^{z-1}, but I'm not seeing at all how to get from point A to point B here.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by davesface View Post
    Obtain the Taylor series e^z=e \sum_{n=0}^{\infty}\frac{(z-1)^n}{n!} (\left |z-1 \right |< \infty) for the function f(z)=e^z by using f^{(n)}(1) (n=0,1,2,...).

    So it would seem to me that f^{(n)}(1)=e, \forall n, but clearly that can't be the case. Clearly that's the correct Taylor series, since it's just e times the Taylor series for e^{z-1}, but I'm not seeing at all how to get from point A to point B here.


    That is a weird way to put things... : f(z)=e^z\Longrightarrow f^{(n)}(z)=e^z\Longrightarrow f^{(n)}(0)=1\,\,\,\forall\,n\in\mathbb{N} , so e^z=1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\  ldots =\sum^\infty_{k=0}\frac{z^k}{k!} \Longrightarrow e^z=e\cdot e^{z-1}=e\sum^\infty_{k=0}\frac{(z-1)^k}{k!} ...weird! Why would anyone want to represent e^z\,\,\,as\,\,\,e\cdot e^{z-1} without given any further reason? Beats me.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Generalized Taylor series of an exponential
    Posted in the Math Challenge Problems Forum
    Replies: 1
    Last Post: March 16th 2011, 12:19 PM
  2. Getting stuck on series - can't develop Taylor series.
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: October 5th 2010, 08:32 AM
  3. Replies: 0
    Last Post: January 26th 2010, 08:06 AM
  4. Formal power series & Taylor series
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 19th 2009, 09:01 AM
  5. Replies: 9
    Last Post: April 3rd 2008, 05:50 PM

Search Tags


/mathhelpforum @mathhelpforum