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Miklós Schweitzer
1967 Miklós Schweitzer
5
Miklos Schweitzer 1967_5
Miklos Schweitzer 1967_5
Source:
October 6, 2008
function
limit
integration
inequalities
geometry
3D geometry
logarithms
Problem Statement
Let
f
f
f
be a continuous function on the unit interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. Show that
lim
n
→
∞
∫
0
1
.
.
.
∫
0
1
f
(
x
1
+
.
.
.
+
x
n
n
)
d
x
1
.
.
.
d
x
n
=
f
(
1
2
)
\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)
n
→
∞
lim
∫
0
1
...
∫
0
1
f
(
n
x
1
+
...
+
x
n
)
d
x
1
...
d
x
n
=
f
(
2
1
)
and
lim
n
→
∞
∫
0
1
.
.
.
∫
0
1
f
(
x
1
.
.
.
x
n
n
)
d
x
1
.
.
.
d
x
n
=
f
(
1
e
)
.
\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).
n
→
∞
lim
∫
0
1
...
∫
0
1
f
(
n
x
1
...
x
n
)
d
x
1
...
d
x
n
=
f
(
e
1
)
.
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