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KoMaL A Problems 2021/2022
A. 810
Infinite Sum Is Zero
Infinite Sum Is Zero
Source: KöMaL A. 810
March 23, 2022
komal
algebra
combinatorics
series
Problem Statement
For all positive integers
n
,
n,
n
,
let
r
n
r_n
r
n
be defined as
r
n
=
∑
i
=
0
n
(
−
1
)
i
(
n
i
)
1
(
i
+
1
)
!
.
r_n=\sum_{i=0}^n(-1)^i\binom{n}{i}\frac{1}{(i+1)!}.
r
n
=
i
=
0
∑
n
(
−
1
)
i
(
i
n
)
(
i
+
1
)!
1
.
Prove that
∑
r
=
1
∞
r
i
=
0.
\sum_{r=1}^\infty r_i=0.
∑
r
=
1
∞
r
i
=
0.
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