The following infinite series can be used to approximate

e^x:
e
x
:

e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}
e
x
=1+x+
2
x
2

​
+
3!
x
3

​
+⋯+
n!
x
n

​


Prove that this Maclaurin series expansion is a special case of the Taylor series expansion with

x_i = 0
x
i
​
=0

and

h = x.
h=x.

Use the Taylor series to estimate

f(x) = e^{-x}
f(x)=e
−x


at

x_{i+1} = 1
x
i+1
​
=1

for

x_i = 0.2
x
i
​
=0.2

Employ the zero-, first-, second-, and third-order versions and compute the

|\varepsilon_t|
∣ε
t
​
∣

for each case.