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2003 IMC
5
IMC 2003 Problem 5
IMC 2003 Problem 5
Source: IMC 2003
March 7, 2021
function
real analysis
Problem Statement
Let
g
:
[
0
,
1
]
→
R
g:[0,1]\rightarrow \mathbb{R}
g
:
[
0
,
1
]
→
R
be a continuous function and let
f
n
:
[
0
,
1
]
→
R
f_{n}:[0,1]\rightarrow \mathbb{R}
f
n
:
[
0
,
1
]
→
R
be a sequence of functions defined by
f
0
(
x
)
=
g
(
x
)
f_{0}(x)=g(x)
f
0
(
x
)
=
g
(
x
)
and
f
n
+
1
(
x
)
=
1
x
∫
0
x
f
n
(
t
)
d
t
.
f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.
f
n
+
1
(
x
)
=
x
1
∫
0
x
f
n
(
t
)
d
t
.
Determine
lim
n
→
∞
f
n
(
x
)
\lim_{n\to \infty}f_{n}(x)
lim
n
→
∞
f
n
(
x
)
for every
x
∈
(
0
,
1
]
x\in (0,1]
x
∈
(
0
,
1
]
.
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