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Vojtěch Jarník IMC
2016 VJIMC
4
Nested infinite sum
Nested infinite sum
Source: VJIMC, 26th annual, Category I, Problem 4
April 10, 2016
real analysis
Vojtech Jarnik
infinite sum
Problem Statement
Find the value of sum
∑
n
=
1
∞
A
n
\sum_{n=1}^\infty A_n
∑
n
=
1
∞
A
n
, where
A
n
=
∑
k
1
=
1
∞
⋯
∑
k
n
=
1
∞
1
k
1
2
1
k
1
2
+
k
2
2
⋯
1
k
1
2
+
⋯
+
k
n
2
.
A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.
A
n
=
k
1
=
1
∑
∞
⋯
k
n
=
1
∑
∞
k
1
2
1
k
1
2
+
k
2
2
1
⋯
k
1
2
+
⋯
+
k
n
2
1
.
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