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,
()
xf
( ) ( ) ( ) ( )
......
1100
++++= xaxaxaxf
nn
ψψψ
∞
→
x
, (1)
...,2,1,0
=
n
( ) ( ) ( ) ( ) ( )( )
xOxaxaxaxf
nnn 11100
......
+
+++++= ψψψψ
, (2)
∞
→
x
.
,
. (3)
()
xf
,
.
4.
( )
dt
t
e
xf
x
t
∫
=
1
+∞
→
x
. (3)
(3)
n
, :
( )
( )
dt
t
e
n
t
n
ttt
exf
x
n
t
x
n
t
∫
+
+
−
++++=
1
1
1
32
!
!1
...
!2!11
1
)1(
limlim
12
1
1
1
1
=
++−
=
−−−−
−−
∞→
−−
−−
∞→
∫
xnxn
xn
x
xn
x
tn
x
exexn
ex
ex
dtet
,
( )
( )
+
−
++++=
+132
1!1
...
!2!11
nn
x
x
O
x
n
xxx
exf
,
∞
→
x
.
( )
...
!
...
!2!11
132
+++++=
+n
x
x
n
x
x
x
exf
(4)
(4)
( )
xf
( )
...
!
...
!2!1
132
+++++=
+
−−−−
n
xxxx
x
en
x
e
x
e
x
e
xf
(5)
(5) .
( )
( )
( )
.11lim
1
!
!1
lim
!
:
!1
lim
2
1
12
>∞=+=
+
=
+
∞→
+
+
∞→
+
−
+
−
∞→
n
xnx
nx
x
en
x
en
n
n
n
n
n
x
n
x
n
, (5)
[
)
+∞;1
,
, (5)
(3).
, f (x )
f ( x ) = a0ψ 0 ( x ) + a1ψ 1 ( x ) + ... + anψ n ( x ) + ... x → ∞, (1)
n = 0,1, 2, ...
f ( x ) = a0ψ 0 ( x ) + a1ψ 1 ( x ) + ... + a nψ n ( x ) + ... + O (ψ n+1 ( x )) , (2)
x → ∞.
,
. (3)
f (x ) ,
.
4.
x
et
f ( x ) = ∫ dt x → +∞ . (3)
1
t
(3) n , :
(n − 1)!
x
t 1
x
et
f ( x) = e + 2 + 3 + ... + n + n! ∫ n+1 dt
1! 2!
t t t t 1 1
t
x
∫t
− n−1 t
e dt
x −n−1e x
lim 1
= lim = 1,
x→∞ x −n−1e x x→∞ − ( n + 1) x − n−2 e x + x − n−1e x
f (x )e x =
1 1! 2! (n − 1)! + O 1
+ 2 + 3 + ... + n+1 , x → ∞.
x x x xn x
f (x)e x =
1 1! 2! n!
+ 2 + 3 + ... + n +1 + ... (4)
x x x x
(4) f (x )
e − x 1!e − x 2!e − x n !e − x
f (x ) = + 2 + 3 + ... + n+1 + ... (5)
x x x x
(5) .
lim
(n + 1)!e − x n !e − x
: = lim
x n+1 (n + 1)! 1
= lim(n + 1) = ∞ > 1.
n →∞ x n+ 2 x n +1 n →∞ x n+ 2 n ! x n→∞
, (5) [1;+∞) ,
, (5)
(3).
