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Problems
Contests
International Contests
Tuymaada Olympiad
1996 Tuymaada Olympiad
6
6
Part of
1996 Tuymaada Olympiad
Problems
(1)
limit of f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)) as n\to \infty
Source: Tuymaada 1996 p6
4/27/2019
Given the sequence
f
1
(
a
)
=
s
i
n
(
0
,
5
π
a
)
f_1(a)=sin(0,5\pi a)
f
1
(
a
)
=
s
in
(
0
,
5
πa
)
f
2
(
a
)
=
s
i
n
(
0
,
5
π
(
s
i
n
(
0
,
5
π
a
)
)
)
f_2(a)=sin(0,5\pi (sin(0,5\pi a)))
f
2
(
a
)
=
s
in
(
0
,
5
π
(
s
in
(
0
,
5
πa
)))
.
.
.
...
...
f
n
(
a
)
=
s
i
n
(
0
,
5
π
(
s
i
n
(
.
.
.
(
s
i
n
(
0
,
5
π
a
)
)
.
.
.
)
)
)
f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))
f
n
(
a
)
=
s
in
(
0
,
5
π
(
s
in
(
...
(
s
in
(
0
,
5
πa
))
...
)))
, where
a
a
a
is any real number. What limit aspire the members of this sequence as
n
→
∞
n \to \infty
n
→
∞
?
algebra
Sequence
Limit of sequence
limit
trigonometric