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2021-IMOC
A9
A nice sum
A nice sum
Source: IMOC 2021 A9
August 11, 2021
Sum
algebra
Problem Statement
For a given positive integer
n
,
n,
n
,
find
∑
k
=
0
n
(
(
n
k
)
⋅
(
−
1
)
k
(
n
+
1
−
k
)
2
−
(
−
1
)
n
(
k
+
1
)
(
n
+
1
)
)
.
\sum_{k=0}^{n} \left(\frac{\binom{n}{k} \cdot (-1)^k}{(n+1-k)^2} - \frac{(-1)^n}{(k+1)(n+1)}\right).
k
=
0
∑
n
(
(
n
+
1
−
k
)
2
(
k
n
)
⋅
(
−
1
)
k
−
(
k
+
1
)
(
n
+
1
)
(
−
1
)
n
)
.
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