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Mathematics
Mathematics
Problem
The function
f
f
f
is defined by
f
(
x
)
=
e
k
x
cos
x
f(x)=e^{kx}\cos x
f
(
x
)
=
e
k
x
cos
x
, where
k
k
k
is a real constant. Its Maclaurin series is
f
(
x
)
=
1
+
k
x
+
k
2
−
1
2
x
2
+
k
3
−
3
k
6
x
3
+
⋯
.
f(x)=1+kx+\frac{k^{2}-1}{2}x^{2}+\frac{k^{3}-3k}{6}x^{3}+\cdots.
f
(
x
)
=
1
+
k
x
+
2
k
2
−
1
x
2
+
6
k
3
−
3
k
x
3
+
⋯
.
Measurements near
x
=
0
x=0
x
=
0
show that the series actually begins
1
+
5
x
+
12
x
2
+
⋯
1+5x+12x^{2}+\cdots
1
+
5
x
+
12
x
2
+
⋯
. Determine the value of
k
k
k
.
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