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IMC
1998 IMC
3
f(x)=2x(1-x)
f(x)=2x(1-x)
Source: IMC 1998 day 1 problem 3
November 1, 2005
limit
integration
real analysis
real analysis unsolved
Problem Statement
Let
f
(
x
)
=
2
x
(
1
−
x
)
,
x
∈
R
f(x)=2x(1-x), x\in\mathbb{R}
f
(
x
)
=
2
x
(
1
−
x
)
,
x
∈
R
and denote
f
n
=
f
∘
f
∘
.
.
.
∘
f
f_n=f\circ f\circ ... \circ f
f
n
=
f
∘
f
∘
...
∘
f
,
n
n
n
times. (a) Find
lim
n
→
∞
∫
0
1
f
n
(
x
)
d
x
\lim_{n\rightarrow\infty} \int^1_0 f_n(x)dx
lim
n
→
∞
∫
0
1
f
n
(
x
)
d
x
. (b) Now compute
∫
0
1
f
n
(
x
)
d
x
\int^1_0 f_n(x)dx
∫
0
1
f
n
(
x
)
d
x
.
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