In this question you may find usefull one of the following Maclaurin expansions e x
=∑ k=0
[infinity]
k!
x k
,sinx=∑ k=0
[infinity]
(−1) k
(2k+1)!
x 2k+1
,cosx=∑ k=0
[infinity]
(−1) k
(2k)!
x 2k
valid for all x∈R, 1−x
1
=∑ k=0
[infinity]
x k
valid for x∈(−1,1) Suppose that the Taylor series for e x
cos(4x) about 0 is a 0
+a 1
x+a 2
x 2
+⋯+a 6
x 6
+⋯ Enter the exact values of a 0
and a 6
in the boxes below. Suppose that a function f has derivatives of all orders at a. Then the series ∑ k=0
[infinity]
k!
f (k)
(a)
(x−a) k
is called the Taylor series for f about a, where f(n) is the nth order derivative of f. Suppose that the Taylor series for e 2x
cos(2x) about 0 is a 0
+a 1
x+a 2
x 2
+⋯+a 4
x 4
+⋯ Enter the exact values of a 0
and a 4
in the boxes below.
