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Nicolae Coculescu
2007 Nicolae Coculescu
3
Another sequence defined by a primitive recursion
Another sequence defined by a primitive recursion
Source:
December 13, 2019
function
real analysis
primitives
Problem Statement
Let
F
:
R
⟶
R
F:\mathbb{R}\longrightarrow\mathbb{R}
F
:
R
⟶
R
be a primitive with
F
(
0
)
=
0
F(0)=0
F
(
0
)
=
0
of the function
f
:
R
⟶
R
f:\mathbb{R}\longrightarrow\mathbb{R}
f
:
R
⟶
R
defined as
f
(
x
)
=
sin
(
x
2
)
,
f(x)=\sin (x^2) ,
f
(
x
)
=
sin
(
x
2
)
,
and let be a sequence
(
a
n
)
n
≥
0
\left( a_n \right)_{n\ge 0}
(
a
n
)
n
≥
0
with
a
0
∈
(
0
,
1
)
a_0\in (0,1)
a
0
∈
(
0
,
1
)
and defined as
a
n
=
a
n
−
1
−
F
(
a
n
−
1
)
.
a_{n}=a_{n-1}-F\left( a_{n-1} \right) .
a
n
=
a
n
−
1
−
F
(
a
n
−
1
)
.
Calculate
lim
n
→
∞
a
n
n
.
\lim_{n\to\infty } a_n\sqrt{n} .
lim
n
→
∞
a
n
n
.
Florian Dumitrel
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