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Putnam
2019 Putnam
B2
Putnam 2019 B2
Putnam 2019 B2
Source:
December 10, 2019
Putnam
Putnam 2019
limit
Problem Statement
For all
n
≥
1
n\ge 1
n
≥
1
, let
a
n
=
∑
k
=
1
n
−
1
sin
(
(
2
k
−
1
)
π
2
n
)
cos
2
(
(
k
−
1
)
π
2
n
)
cos
2
(
k
π
2
n
)
a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}
a
n
=
∑
k
=
1
n
−
1
c
o
s
2
(
2
n
(
k
−
1
)
π
)
c
o
s
2
(
2
n
kπ
)
s
i
n
(
2
n
(
2
k
−
1
)
π
)
. Determine
lim
n
→
∞
a
n
n
3
\lim_{n\rightarrow \infty}\frac{a_n}{n^3}
lim
n
→
∞
n
3
a
n
.
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