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Mathematics
Mathematics
Problem
Determine the elementary function represented by the power series
∑
n
=
0
∞
(
−
1
)
n
(
x
2
)
n
n
!
2
n
\displaystyle \sum_{n=0}^{\infty}\dfrac{(-1)^n (x^{2})^{n}}{n!\,2^{n}}
n
=
0
∑
∞
n
!
2
n
(
−
1
)
n
(
x
2
)
n
for all real
x
x
x
. a)
e
x
2
/
2
e^{\,x^{2}/2}
e
x
2
/2
b)
e
−
x
2
/
2
e^{-x^{2}/2}
e
−
x
2
/2
c)
cos
(
x
2
2
)
\cos\!\left(\dfrac{x^{2}}{2}\right)
cos
(
2
x
2
)
d)
e
−
x
2
e^{-x^{2}}
e
−
x
2
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