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2006 Stanford Mathematics Tournament
15
2006 SMT Team Round #15 - Computation with Geometric Series
2006 SMT Team Round #15 - Computation with Geometric Series
Source:
August 22, 2011
limit
geometric series
Problem Statement
Let
c
i
c_i
c
i
denote the
i
i
i
th composite integer so that
{
c
i
}
=
4
,
6
,
8
,
9
,
.
.
.
\{c_i\}=4,6,8,9,...
{
c
i
}
=
4
,
6
,
8
,
9
,
...
Compute
∏
i
=
1
∞
c
i
2
c
i
2
−
1
\prod_{i=1}^{\infty} \dfrac{c^{2}_{i}}{c_{i}^{2}-1}
i
=
1
∏
∞
c
i
2
−
1
c
i
2
(Hint:
∑
n
=
1
∞
1
n
2
=
π
2
6
\textstyle\sum^\infty_{n=1} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}
∑
n
=
1
∞
n
2
1
=
6
π
2
)
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