Difference between revisions of "Taylor series of the exponential function"

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==Theorem==
 
==Theorem==
Let $z_0 \in \mathbb{C}$. The following [[Taylor series]] holds for all $z \in \mathbb{C}$:
+
The following [[Taylor series]] holds for all $z \in \mathbb{C}$:
$$e^{z} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(z-z_0)^k}{k!},$$
+
$$e^{z} = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!},$$
 
where $e^z$ is the [[exponential function]].
 
where $e^z$ is the [[exponential function]].
  

Latest revision as of 04:03, 3 October 2016

Theorem

The following Taylor series holds for all $z \in \mathbb{C}$: $$e^{z} = \displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{k!},$$ where $e^z$ is the exponential function.

Proof

References