How do you find the Taylor series of $f (x) = e^{x}$ $f(x)=e^x$ ?

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Answer 1 · 2014-09-21T16:54:14

Taylor series at $x = 0$ $x=0$ (also called Maclaurin series) for $f (x)$ $f(x)$ is

$f (x) = \infty \sum n = 0 \frac{f^{(n)} (0)}{n!} x^{n}$ $f(x)=sum_{n=0}^infty{f^{(n)}(0)}/{n!}x^n$ .

Since if $f (x) = e^{x}$ $f(x)=e^x$ , then

$f(x)=f'(x)=f''(x)=cdots=f^{(n)}(x)=e^x$ ,

so,

$f(0)=f'(0)=f''(0)=cdots=f^{(n)}(0)=e^0=1$

Hence, the Maclaurin series is

$f(x)=sum_{n=0}^infty 1/{n!}x^n=sum_{n=0}^infty x^n/{n!}$

How do you find the Taylor series of f(x)=e^xf(x)=exf(x)=e^x ?