We have found the following derivatives of f(x). f(x)
f ′
(x)
f ′′
(x)
f ′′′
(x)
f (4)
(x)
​
=8(1−x) −2
=16(1−x) −3
=48(1−x) −4
=192(1−x) −5
=960(1−x) −6
​
The Maclaurin series uses the function and its derivatives evaluated at 0 , which we find as follows. f(0)=
f ′
(0)=
f ′′
(0)=
f ′′′
(0)=
f (4)
(0)=
​
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R n
​
(x)→0.] f(x)=e −4x
f(x)=∑ n=0
[infinity]
​
(
​
Find the associated radius of convergence R. R= Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that R n
​
(x)→0.] x 5
+4x 3
+x,a=3 f(x)= Find the associated radius of convergence R. R=