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Putnam
1967 Putnam
B3
Putnam 1967 B3
Putnam 1967 B3
Source: Putnam 1967
May 14, 2022
Putnam
function
periodic function
Integral
Problem Statement
If
f
f
f
and
g
g
g
are continuous and periodic functions with period
1
1
1
on the real line, then
lim
n
→
∞
∫
0
1
f
(
x
)
g
(
n
x
)
d
x
=
(
∫
0
1
f
(
x
)
d
x
)
(
∫
0
1
g
(
x
)
d
x
)
.
\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).
n
→
∞
lim
∫
0
1
f
(
x
)
g
(
n
x
)
d
x
=
(
∫
0
1
f
(
x
)
d
x
)
(
∫
0
1
g
(
x
)
d
x
)
.
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