Find the series for $e^{-x}$ (By differentiation of $e^x$)

Question

Given that $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$

Find the series for $e^{-x}$

Note: derive the series for $e^{-x}$ showing your own working.

Trying to work out some questions from my textbook, I don't even know where to start. Help

Answer 1

One thing that will help you out a lot:

If $f(x)=e^x$, and $f(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}...$

Then $f(-x)=e^{-x}$. How does that affect the power series?

Answer 2

Hint: Substitute $-x$ instead of $x$.