What is the Taylor series of e(−x)2?
Answers
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0
The answer, when
a
=
0
, is :
f
(
x
)
=
∞
∑
k
=
0
x
2
k
k
!
The Taylor series is given by :
f
(
x
)
=
∞
∑
k
=
0
f
(
k
)
(
a
)
k
!
(
x
−
a
)
k
.
We know that the Taylor series of
e
x
, when
a
=
0
, is :
f
(
x
)
=
∞
∑
k
=
0
x
k
k
!
So now, we just need to replace the
x
of the above series with
(
−
x
)
2
(in operations with Taylor series, it is called substitution) :
f
(
x
)
=
∞
∑
k
=
0
(
(
−
x
)
2
)
k
k
!
=
∞
∑
k
=
0
(
−
x
)
2
k
k
!
=
∞
∑
k
=
0
x
2
k
k
!
If you meant
e
−
(
x
2
)
, it would be :
f
(
x
)
=
∞
∑
k
=
0
(
−
x
2
)
k
k
!
=
∞
∑
k
=
0
(
−
1
)
k
⋅
x
2
k
k
!
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