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Vojtěch Jarník IMC
2011 VJIMC
Problem 3
infinite sums equal
infinite sums equal
Source: VJIMC 2011 1.3
June 1, 2021
Summation
Problem Statement
Prove that
∑
k
=
0
∞
x
k
1
+
x
2
k
+
2
(
1
−
x
2
k
+
2
)
2
=
∑
k
=
0
∞
(
−
1
)
k
x
k
(
1
−
x
k
+
1
)
2
\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}
k
=
0
∑
∞
x
k
(
1
−
x
2
k
+
2
)
2
1
+
x
2
k
+
2
=
k
=
0
∑
∞
(
−
1
)
k
(
1
−
x
k
+
1
)
2
x
k
for all
x
∈
(
−
1
,
1
)
x\in(-1,1)
x
∈
(
−
1
,
1
)
.
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