Question

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

Answer 1

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

`f(x)=sum_{n=0}^infty{f^{(n)}(0)}/{n!}x^n`.

Since if `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!}`