Taylor Series centred at some value.

Question

The Taylor Series for $\bf{e}^x$ centred at $x=0$ is:$${{\bf{e}}^x} = \sum\limits_{n = 0}^\infty {\frac{{{x^n}}}{{n!}}}$$

Now lets say I let $x=100$, will the Taylor Series above give me the correct value for $\bf{e^{100}}$ since I have centered it at $x=0$ ?

Does centering the Taylor series at any number limit me in any way?

Answer 1

In the case of the exponential function, the answer is negative. It turns out that, for every $x\in\mathbb R$, $e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$.