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IMC
2002 IMC
9
9
Part of
2002 IMC
Problems
(1)
IMC 2002 Problem 9
Source: IMC 2002
3/7/2021
For each
n
≥
1
n\geq 1
n
≥
1
let
a
n
=
∑
k
=
0
∞
k
n
k
!
,
b
n
=
∑
k
=
0
∞
(
−
1
)
k
k
n
k
!
.
a_{n}=\sum_{k=0}^{\infty}\frac{k^{n}}{k!}, \;\; b_{n}=\sum_{k=0}^{\infty}(-1)^{k}\frac{k^{n}}{k!}.
a
n
=
k
=
0
∑
∞
k
!
k
n
,
b
n
=
k
=
0
∑
∞
(
−
1
)
k
k
!
k
n
.
Show that
a
n
⋅
b
n
a_{n}\cdot b_{n}
a
n
⋅
b
n
is an integer.
real analysis
Summation