1. ## Taylor Series, Complex

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

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

2. Originally Posted by eskimo343
Obtain the Taylor series $\displaystyle e^z=e \cdot \sum_{n=0}^{\infty} \frac{(z-1)^n}{n!}$ ($\displaystyle |z - 1 | < \infty$) for the function $\displaystyle f(z)=e^z$ by using $\displaystyle f^{(n)}(1)$ where $\displaystyle (n=1, 2, \ldots)$.

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

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