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.
