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Putnam
2020 Putnam
A6
A6
Part of
2020 Putnam
Problems
(1)
Putnam 2020 A6
Source: 81st William Lowell Putnam Competition
2/22/2021
For a positive integer
N
N
N
, let
f
N
f_N
f
N
be the function defined by
f
N
(
x
)
=
∑
n
=
0
N
N
+
1
/
2
−
n
(
N
+
1
)
(
2
n
+
1
)
sin
(
(
2
n
+
1
)
x
)
.
f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right).
f
N
(
x
)
=
n
=
0
∑
N
(
N
+
1
)
(
2
n
+
1
)
N
+
1/2
−
n
sin
(
(
2
n
+
1
)
x
)
.
Determine the smallest constant
M
M
M
such that
f
N
(
x
)
≤
M
f_N (x)\le M
f
N
(
x
)
≤
M
for all
N
N
N
and all real
x
x
x
.
Putnam
Putnam 2020