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Файл:Числовые и функциональные ряды. Учебник
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Ⱦɥɹ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ ɜɢɞɚ (5.1) ɜɫɟ ɫɤɚɡɚɧɧɨɟ ɜɵɲɟ ɨɫɬɚɟɬɫɹ ɜ ɫɢɥɟ ɫ ɬɨɣ
x
ɬɨɥɶɤɨ ɪɚɡɧɢɰɟɣ, ɱɬɨ ɬɟɩɟɪɶ ɰɟɧɬɪ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ ɛɭɞɟɬ ɥɟɠɚɬɶ ɧɟ ɜ ɬɨɱɤɟ
0=
, ɚ ɜ ɬɨɱɤɟ
()
RxRx +−00,
.xx=
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɧɬɟɪɜɚɥɨɦ ɫɯɨɞɢɦɨɫɬɢ ɛɭɞɟɬ ɢɧɬɟɪɜɚɥ
0
.
5.2. ȼɵɱɢɫɥɟɧɢɟ ɪɚɞɢɭɫɚ ɢ ɢɫɫɥɟɞɨɜɚɧɢɟ ɨɛɥɚɫɬɢ
ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ
ɋɨɫɬɚɜɢɦ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ (5.2):
10
Ʉ ɪɹɞɭ (5.3), ɱɥɟɧɵ ɤɨɬɨɪɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɵ, ɩɪɢɦɟɧɢɦ ɩɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ.
ȼɵɱɢɫɥɢɦ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɩɨɫɥɟɞɭɸɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ (5.3) ɤ ɩɪɟɞɵɞɭɳɟɦɭ.
ȼ ɨɬɥɢɱɢɟ ɨɬ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ, ɷɬɨɬ ɩɪɟɞɟɥ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ ɦɧɨɠɢɬɟɥɟɦ
n
1
+
⋅
xa
n
1
+
n
∞→
⋅
xa
n
x
n
n
∞→
Ⱦɥɹ ɬɟɯ ɡɧɚɱɟɧɢɣ
x
, ɩɪɢ ɤɨɬɨɪɵɯ ɩɨɥɭɱɚɟɦɵɣ ɩɪɟɞɟɥ ɛɭɞɟɬ ɦɟɧɶɲɟ 1, ɪɹɞ
ɫɯɨɞɢɬɫɹ, ɚ ɞɥɹ ɬɟɯ, ɩɪɢ ɤɨɬɨɪɵɯ ɛɨɥɶɲɟ 1, ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ. Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ,
ɬɨ ɡɧɚɱɟɧɢɟ
ɭɫɨɦ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ, ɬ. ɟ.
x
, ɩɪɢ ɤɨɬɨɪɨɦ ɷɬɨɬ ɩɪɟɞɟɥ ɪɚɜɟɧ ɟɞɢɧɢɰɟ, ɢ ɛɭɞɟɬ ɹɜɥɹɬɶɫɹ ɪɚɞɢ-
+
1
a
n
lim =
⋅
R
a
n
∞→
n
a
lim
=
R
a
n
∞→
n
Ⱥɧɚɥɨɝɢɱɧɨ, ɞɥɹ ɪɹɞɚ (5.3) ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɪɚɞɢɤɚɥɶɧɵɦ ɩɪɢɡɧɚɤɨɦ
Ʉɨɲɢ ɢ ɜɵɱɢɫɥɢɬɶ ɩɪɟɞɟɥ
n
n
n
n
Ɍɨɝɞɚ, ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɧɚɣɞɟɦ ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɟɧɫɬɜɚ ɷɬɨɝɨ ɩɪɟɞɟɥɚ ɟɞɢ-
ɧɢɰɟ, ɬ. ɟ.
n
∞→
n
1lim =⋅
aR
,
n
R
1
=
n
lim
n
∞→
n
......
+⋅++⋅+
xaxaa
n
a
n
+
a
n
1
,
n
. (5.4)
1
+
⋅=⋅ limlim
n
. (5.3)
x
:
a
n
11
+
limlimlim
⋅=⋅=
x
n
∞→
n
.
axxa
n
∞→∞→
.
a
n
. (5.5)
a
n
81
()
x
x
x
x
ȿɫɥɢ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɪɚɜɟɧ ɛɟɫɤɨɧɟɱɧɨɫɬɢ
ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ
ȿɫɥɢ ɩɪɢ ɜɫɟɯ
ɜɫɸɞɭ ɪɚɫɯɨɞɢɬɶɫɹ (ɤɪɨɦɟ ɬɨɱɤɢ 0=
.
0≠x
ɩɪɟɞɟɥ ɨɤɚɠɟɬɫɹ ɪɚɜɧɵɦ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɬɨ ɪɹɞ ɛɭɞɟɬ
) ɢ ɟɝɨ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɛɭɞɟɬ ɪɚɜɟɧ
∞=R
, ɬɨ ɪɹɞ (5.2) ɫɯɨɞɢɬɫɹ
ɧɭɥɸ.
Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ.
ɉɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɥɸɛɨɝɨ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ ɧɟɨɛɯɨ-
ɞɢɦɨ:
1) ɜɵɱɢɫɥɢɬɶ ɩɨ ɮɨɪɦɭɥɚɦ (5.4) ɢɥɢ (5.5) ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ;
2) ɡɚɩɢɫɚɬɶ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ;
3) ɩɪɨɜɟɪɢɬɶ ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ ɧɚ ɤɚɠɞɨɣ ɝɪɚɧɢɰɟ ɢɧɬɟɪɜɚɥɚ.
ɉɪɢɦɟɪ 5.1.
Ɉɩɪɟɞɟɥɢɬɶ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ:
∞+
n
x
1)
2)
3)
4)
;
¦
!
n
=1
n
∞+
n
x
;
¦
n
=1n
∞+
n
x
¦
n
()
+133
n
=
n
∞+
n
()( )
¦
()()
=
1
n
;
n
+⋅+
713
xn
nn
.
+⋅+
65
Ɋɟɲɟɧɢɟ.
1. ɂɦɟɟɦ
=
a
,
n
a
+
n
!1n
1
=
1
() ()
+
1
=
!1
. ȼɵɱɢɫɥɢɦ, ɢɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ
1!
+⋅
nnn
(5.4), ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ:
()
+⋅
1!
=
lim n
R
nn
!
n
()
∞→∞→
nn
1lim
.
+∞=+=
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɹɞ ɫɯɨɞɢɬɫɹ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ.
2. ɂɦɟɟɦ
a
a
=
,
n
n
n
1
Ɍɨɝɞɚ, ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ:
1
=
1
+
. ɉɨ ɮɨɪɦɭɥɟ (5.4) ɩɨɥɭɱɚɟɦ
1
+
n
11 <<−
. ɉɪɨɜɟɪɢɦ ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ ɧɚ ɤɨɧ-
R
=
n
lim =
ɰɚɯ ɢɧɬɟɪɜɚɥɚ.
∞+
n
()
ɉɪɢ
1−=
, ɩɨɥɭɱɚɟɦ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ
¦
n
−11
, ɤɨɬɨɪɵɣ ɫɯɨɞɢɬɫɹ
n
=
ɭɫɥɨɜɧɨ (ɫɦ. ɩɪɢɦɟɪ 3.3).
n
1
+
1
.
n
∞→
82
x
∞+
x
x
x
x
x
ɉɪɢ
1=
, ɩɨɥɭɱɚɟɦ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ
1
, ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ ɪɚɫɯɨɞɹ-
¦
n
=1
n
ɳɢɦɫɹ.
ɂɬɚɤ, ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ – ɷɬɨ ɢɧɬɟɪɜɚɥ
ɢɧɬɟɪɜɚɥɚ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ
3. ɂɦɟɟɦ
a
n
n
()
1−=
, ɪɹɞ ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ.
,
a
+
1
n
331+=n
=
1
+
n
1
()
+
n
ɇɚɣɞɟɦ ɩɨ ɮɨɪɦɭɥɟ (5.4) ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
33 <<−
ɬɨɝɞɚ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ:
.
ɂɫɫɥɟɞɭɟɦ ɫɯɨɞɢɦɨɫɬɶ ɧɚ ɝɪɚɧɢɰɚɯ, ɬ. ɟ. ɩɪɢ
∞
n
=
¦¦
+
33
n
()
−
n
=
11
n
1
.
+
3
ȿɫɥɢ 3−=
, ɬɨ
∞
=
n
()
n
−
3
()
n
.
43
, ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ ɷɬɨɝɨ
[
)
1;1−
R
lim
=
n
∞→
3−=
ɢ 3=x.
n
1
+
()
n
()
n
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɢɡɧɚɤɨɦ Ʌɟɣɛɧɢɰɚ:
a
n
,
31+=n
...21>> aa
a
,
n
=
n
1
limlim =
+
n
∞→∞→
n
0
.
3
ɉɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ.
∞
Ɋɚɫɫɦɨɬɪɢɦ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ:
¦
n
1
.
+13
n
=
Ⱦɚɧɧɵɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɪɚɫɯɨɞɹɳɢɦɫɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɹɞ ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ.
∞
3=
ȿɫɥɢ
, ɬɨ
=
Ɋɹɞ ɫɯɨɞɢɬɫɹ ɩɪɢ
4. ɂɦɟɟɦ
=
a
n
()()
n
3
n
n
[
ɩ
()
+⋅
n
ɩɩ
∞
1
=
¦¦
+
)3(3
=
nn
.
)
3;3−∈x
,
a
6513+⋅+
– ɪɚɫɯɨɞɢɬɫɹ.
+
3
n
11
1
+
ɩ
=
+
1
n
()()
()
23
+⋅
n
, ɬɨɝɞɚ ɩɨ ɮɨɪɦɭɥɟ (5.4)
76
+⋅+
ɩɩ
ɢɦɟɟɦ
n
=
lim
R
n
()( )( )
1
+
n
∞→
()()()
+⋅+⋅+⋅
nnn
nnn
1
7613
=
.
3
5623
+⋅+⋅+⋅
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɢɦɟɟɬ ɜɢɞ:
1
§
7x
¨
3
©
1
·
ɢɥɢ
7;
+−−−∈
¸
3
¹
22
20
§
¨
3
©
·
.
;
−−
¸
3
¹
n
43
+
,3
=
33
+
83
ɂɫɫɥɟɞɭɟɦ ɫɯɨɞɢɦɨɫɬɶ ɧɚ ɝɪɚɧɢɰɚɯ ɨɛɥɚɫɬɢ, ɬ. ɟ. ɜ ɬɨɱɤɚɯ
ȿɫɥɢ
1
7 −−=x
ɢ
3
ɩ
∞
1
7 +−=x
, ɬɨ
3
=
§
()
n
()() ()()
713
¨
©
ɩɩ
1
7 +−=x
.
3
n
1
·
++−⋅+⋅
7
¸
3
+⋅+
65
∞
¹
=
¦¦
11
nn
=
+
1
n
+⋅+
65
ɩɩ
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɢɧɬɟɝɪɚɥɶɧɵɦ ɩɪɢɡɧɚɤɨɦ.
∞+ b
()() ()()
1
+
x
65
+⋅+
xx
dx
lim
=
³³
∞→
b
11
45
−+
x
+⋅+
xx
=
dx
65
§
¨
bb
1
¨
=
lim
¨
³³
x
∞→
b
¨
11
¨
©
−
4
dx
+
6
1
2
11
·
§
+
x
¸
¨
2
¹
©
·
¸
¸
=
dx
¸
1
¸
−
¸
4
¹
b
+
x
5
§
¨
b
©
−+=
x
ln166lnlim
·
=
¸
+
x
6
¹
1
lnlim
∞→∞→
b
()
x
()
x
b
17
+
6
+
5
=
16
1
=
¨
∞→
b
©
§
¨
lnlim
17
()
6
+
b
16
()
b
5
+
17
·
7
¸
ln
−
16
6
.
∞=
¸
¹
ɂɧɬɟɝɪɚɥ ɪɚɫɯɨɞɢɬɫɹ, ɡɧɚɱɢɬ ɢ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ.
1
7 −−=x
ɉɪɢ
, ɢɦɟɟɦ
3
n
1
∞
=
n
ɩ
§
()
n
()()
713
¨
©
ɩɩ
·
+−−⋅+⋅
7
¸
3
+⋅+
65
∞
¹
=
¦¦
=
n
n
()
()()
11
+⋅−
11
n
.
+⋅+
65
ɩɩ
Ɋɹɞ ɫɯɨɞɢɬɫɹ ɩɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ. ɇɨ ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɪɹɞ ɢɡ ɚɛɫɨɥɸɬ-
22
1
22
7 −=−−=x
3
20
·
. X
−−∈
;
¸
3
3
¹
ɪɹɞ ɫɯɨɞɢɬɫɹ
3
ɧɵɯ ɜɟɥɢɱɢɧ ɪɚɫɯɨɞɢɬɫɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ
ɭɫɥɨɜɧɨ. ɂɬɚɤ, ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
ª
x
«
¬
.
84
5.3. ɋɜɨɣɫɬɜɚ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ
ɉɟɪɟɱɢɫɥɢɦ ɫɜɨɣɫɬɜɚ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ, ɧɟɨɛɯɨɞɢɦɵɟ ɞɥɹ ɢɯ ɞɚɥɶɧɟɣɲɟɝɨ
ɢɡɭɱɟɧɢɹ ɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ.
1. ɉɭɫɬɶ ɫɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɢɦɟɟɬ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ
()
. Ɍɨɝɞɚ
RR,−
ɪɹɞɵ, ɩɨɥɭɱɟɧɧɵɟ ɢɡ ɧɟɝɨ ɩɨɱɥɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ,
ɢɦɟɸɬ ɬɨɬ ɠɟ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ, ɱɬɨ ɢ ɞɚɧɧɵɣ ɪɹɞ.
.
; ɚ r – ɩɪɨ-
RR,−
2. ɉɭɫɬɶ ɫɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɢɦɟɟɬ ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ
ɢɡɜɨɥɶɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ, ɦɟɧɶɲɟɟ ɱɟɦ
()
ɩɟɧɧɨɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ ɧɚ ɨɬɪɟɡɤɟ
()
. Ɍɨɝɞɚ ɞɚɧɧɵɣ ɫɬɟ-
RrR <<0
[]
rr,−
3. ɋɭɦɦɚ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ (5.2) ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɜ ɤɚɠɞɨɣ
ɬɨɱɤɟ ɟɝɨ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ
()
.
RR,−
4. ɋɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ ɜ ɥɸɛɨɣ ɬɨɱɤɟ
ɟɝɨ ɢɧɬɟɪɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ
′
()
n
++++
n
2110
1
−n
......2......
++++=
xnaxaaxaxaa
n
.
5. ɋɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɜ ɢɧɬɟɪɜɚɥɟ ɫɯɨɞɢ-
ɦɨɫɬɢ
()
, ɬ. ɟ. ɟɫɥɢ
RR,−
ɢ
x
– ɬɨɱɤɢ, ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɢɧɬɟɪɜɚɥɭ ɫɯɨɞɢɦɨ-
x
1
2
ɫɬɢ, ɬɨ
x
2
()
x
1
n
n
x
2
x
1
x
2
1010
x
1
x
2
n
n
³³³³
x
1
.
............
++++=⋅++++
dxxadxxadxadxxaxaa
85
ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
x
ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɞɚɧɧɵɯ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ ɩɨ ɫɬɟɩɟɧɹɦ
5.1.
∞
ɩ
ɯ
1)
3)
5)
7)
¦
n
¦
n
¦
n
¦
n
=
∞
=
∞
=0
∞
=0
; 2)
+01
ɩ
ɩ
ɯ
()
3
; 4)
−1!12
ɩ
2
ɩ
ɯɩ
; 6)
ɩ
2
ɩɩ
ɯ
; 8)
!
ɩ
5.2. ɇɚɣɬɢ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɞɚɧɧɵɯ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ ɩɨ ɫɬɟɩɟɧɹɦ
()
:
xx −
0
∞
()
1)
3)
5)
7)
9)
10)
12
ɯ
¦
ɩ
=
0
n
∞
()
ɯɩ
¦
2
ɩ
=
1
n
∞
()
−
ɯ
¦
()
ɩ
=
1
n
∞
ɩ
¦
()()
=
1
n
2
+
1
2
⋅
∞
()
1
¦
=
1
n
∞
()
xn
¦
=1n
ɩ
−
1
()
+⋅
12
ɩ
ɩ
−⋅
5
+
1
ɩ
2
; 6)
+
1ln
()
−⋅
72
ɯɩ
+⋅+
ɩɩ
+
33
−
1
n
⋅−
n
.
∞
¦
=
n
∞
¦
=
0
n
∞
¦
=0
n
∞
¦
=1
n
; 2)
; 4)
ɩ
; 8)
32
2
4
+
22
35
⋅
−
12
ɩ
x
; 11)
−
12
n
ɩ
ɯ
;
()
+13
ɩɩ
ɩ
+
12
()
ɩɯ
5
4
8
ɩ
ɯ
−
1
()
ɩ
ɩ
;
ɩ
ɩɩ
ɯ
!
ɩ
∞
¦
=
0
n
∞
¦
=
1
n
∞
¦
=
1
n
∞
¦
=
1
n
3
xxx
32
37
⋅
∞
¦
=
n
;
+
!12
.
ɩ
()
ɯ
−
2
()
ɩ
2
ɩ
⋅+
212
ɩ
()
ɯɩ
+
4
3
+
1
ɩ
()
−
2
ɯ
()()
ɩ
()
+
32
ɯ
()( )
+
()
+⋅+
ɩɩ
...
;
ɩ
x
⋅−1212
;
;
ɩ
;
+⋅+
1ln1
ɩɩ
ɩ
;
21
;
nn
:
86
Ƚɥɚɜɚ 6
x
ɊȺɁɅɈɀȿɇɂȿ ɎɍɇɄɐɂɃ
ȼ ɋɌȿɉȿɇɇɕȿ ɊəȾɕ
Ʉɚɤ ɛɵɥɨ ɭɫɬɚɧɨɜɥɟɧɨ ɪɚɧɟɟ, ɫɭɦɦɚ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ
ɹɜɥɹɟɬɫɹ ɧɟɩɪɟ-
()
xS
ɪɵɜɧɨɣ ɢ ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɪɚɡ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɮɭɧɤɰɢɟɣ ɜ ɢɧɬɟɪɜɚɥɟ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ. Ɋɚɫɫɦɨɬɪɢɦ ɜɨɡɦɨɠɧɨɫɬɶ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɡɚɞɚɧɧɨɣ ɮɭɧɤɰɢɢ
ɜ ɜɢɞɟ ɫɭɦɦɵ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ, ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɬɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ
ɧɢɟɦ ɮɭɧɤɰɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ.
()
xf
ɪɚɡɥɨɠɟ-
6.1. Ɋɹɞɵ Ɍɟɣɥɨɪɚ ɢ Ɇɚɤɥɨɪɟɧɚ
ɉɭɫɬɶ ɮɭɧɤɰɢɹ
() ()() ()
ɢɧɬɟɪɜɚɥ ɫɯɨɞɢɦɨɫɬɢ ɤɨɬɨɪɨɝɨ
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɨɜɨɪɹɬ, ɱɬɨ ɮɭɧɤɰɢɹ
ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
ɇɚɣɞɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ
ɂɡɜɟɫɬɧɨ, ɱɬɨ ɜ ɢɧɬɟɪɜɚɥɟ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɬɶ, ɩɪɢɱɟɦ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɬɫɹ ɪɹɞ, ɢɦɟɸɳɢɣ ɬɨɬ ɠɟ ɢɧɬɟɪɜɚɥ
ɫɯɨɞɢɦɨɫɬɢ
()
ɪɭɟɦ ɪɚɜɟɧɫɬɜɨ (6.1), ɩɨɥɭɱɚɟɦ ɬɨɠɞɟɫɬɜɚ, ɫɩɪɚɜɟɞɥɢɜɵɟ ɞɥɹ ɥɸɛɨɝɨ
ɜɚɥɚ ɫɯɨɞɢɦɨɫɬɢ:
() ()()()()
′
() ()() ()
()( )( ) ()()( )
ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ (6.1), ɬ. ɟ.
()
xf
2
02010
()
ɢɥɢ ɩɨ ɫɬɟɩɟɧɹɦ
x
0
10 n
, ɱɬɨ ɢ ɢɫɯɨɞɧɵɣ ɪɹɞ. ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɢɮɮɟɪɟɧɰɢ-
RaRa +− ,
n
() ()
n
RaRa +− ,
()
xf
.xx−
0
ɷɬɨɝɨ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ.
...,...,,,
aaa
2
02010
n
n
.
ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɜ
3
03
1
++n
+−+−+
xxaxxa
010
n
......
+−++−+−+=
xxaxxaxxaaxf
, (6.1)
0
4
xxaxxaxxaxxaaxf
04
,......
2
03021
1
− n
()()()
n
n
0
n
xxanxxna
+
01
3
...432
+−+−+−+=
xxaxxaxxaaxf
04
,...1...
+−++−+
′′
() () ()
2
− n
()()()()
n
n
0
+
n
2
...43322
+−⋅+−⋅+=
xxaxxaaxf
04032
1
−
,...11...
+−++−−+
xxnanxxann
01
′′′
() ( )
3
− n
n
0
n
...43232
+−⋅⋅+⋅=
xxaaxf
043
2
−
xxannnxxannn
01
+
n
...
+−+−+−+−+=
,...1121
+−−++−−⋅−+
ɢɡ ɢɧɬɟɪ-
87
"""""""
n
() ()( ) ()() ( )
+
nn
...,23...1123...21
+−⋅−++⋅−⋅−⋅=
xxannnannnxf
01
"""""""
ɉɪɢ
ɢɦɟɟɦ
xx =
0
() () () ()
=
′
=
′′
=
′′′
,32,2,,
axfaxfaxfaxf ⋅=
30201000
()
n
() ( )( )
0 n
...,23...21
annnxf ⋅−⋅−⋅=
Ɍɨɝɞɚ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ:
′′
()
xf
() ()
′′′
()
xf
=
a
3
32
⋅
′
==
...,,
,,
axfaxfa
20100
()
n
a
=
n
=
2
()
xf
00
...432
n
⋅⋅
0
,
...,
ɢɥɢ
a
2
()
n
=
n
′′
()
xf
0
!2
()
xf
00
!
,
....,
a
3
()
=
,
axfa
100
′′′
()
xf
!3
==
...,,
′
()
xf
0
,
!1
=
a
n
ɉɨɞɫɬɚɜɢɦ ɧɚɣɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɪɚɜɟɧɫɬɜɨ (6.1), ɩɨɥɭɱɚɟɦ
′′′
fx f x
()
fx fx x x x x
=+ −+ −+
()
()
00 0
00
()
1! 2!
()
()
2
′′′
fx f x
()
00
+−++ −+=
3! !
3
()
..... ...
00
n
()
+∞
fx
¦
n
=
0
()
n
!
=−
ɂɬɚɤ, ɟɫɥɢ ɮɭɧɤɰɢɹ
ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
()
xf
ɬɨ ɷɬɨɬ ɪɹɞ ɢɦɟɟɬ ɜɢɞ (6.2) ɢ ɧɚɡɵɜɚɟɬɫɹ
n
()
()
()
n
0
xx
()
0
n
(6.2)
.
ɪɹɞɨɦ Ɍɟɣɥɨɪɚ ɞɥɹ ɮɭɧɤɰɢɢ
88
.
()
xf
,xx−
0
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɩɪɢ
ɪɹɞ (6.2) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
00=x
()
′
()
0
()
f
f
+
0
!1
′′
()
0
f
+
x
2
x
!2
n
()
0
f
...
n
x
!
n
∞+
()
n
()
0
f
=+++
...
¦
=
0
n
n
. (6.3)
x
!
n
ɗɬɨɬ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ
ɪɹɞɨɦ Ɇɚɤɥɨɪɟɧɚ ɞɥɹ ɮɭɧɤɰɢɢ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ ɮɭɧɤɰɢɹ ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
ɪɹɞ ɹɜɥɹɟɬɫɹ ɟɟ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ (ɢɥɢ ɪɹɞɨɦ Ɇɚɤɥɨɪɟɧɚ, ɟɫɥɢ
.
()
xf
,xx−
ɬɨ ɷɬɨɬ
0
).
00=x
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɮɭɧɤɰɢɹ ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
ɬɨ ɨɧɚ ɢɦɟɟɬ ɩɪɨɢɡɜɨɞɧɵɟ ɜɫɟɯ ɩɨɪɹɞɤɨɜ ɜ ɬɨɱɤɟ
,xx−
0
ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ ɜ ɬɨɱɤɟ
x
0
.
ɢɥɢ, ɤɚɤ ɝɨɜɨɪɹɬ,
,xx−
0
Ɋɚɫɫɦɨɬɪɢɦ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ. ɉɭɫɬɶ ɞɚɧɚ ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɜ
ɬɨɱɤɟ
ɮɭɧɤɰɢɹ
x
0
. ɋɨɫɬɚɜɢɦ ɞɥɹ ɧɟɟ ɮɨɪɦɚɥɶɧɨ ɪɹɞ Ɍɟɣɥɨɪɚ
()
xf
()
n
()
xf
...
0
!
n
n
()
...
+−⋅++−⋅
xx
0
+
′′
()
xfxf
00
()
!2!1
xx
2
0
′
0
+
()
()
xf .
Ɉɩɪɟɞɟɥɢɦ, ɩɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɫɭɦɦɚ ɪɹɞɚ Ɍɟɣɥɨɪɚ
ɫɨɜɩɚɞɚɟɬ ɫ ɮɭɧɤ-
()
xS
ɰɢɟɣ, ɞɥɹ ɤɨɬɨɪɨɣ ɷɬɨɬ ɪɹɞ ɫɨɫɬɚɜɥɟɧ. Ɂɚɩɢɲɟɦ ɱɚɫɬɢɱɧɭɸ ɫɭɦɦɭ ɪɹɞɚ Ɍɟɣɥɨɪɚ
()
() ( )
n
xfxS −⋅++−⋅
′
()
xf
0
+= (6.4)
0
()
′′
()
xf
xx
0
0
+−⋅
!2!1
2
()
xx
0
....
n
()
xf
0
()
!
n
n
.
xx
0
ɗɬɚ ɱɚɫɬɢɱɧɚɹ ɫɭɦɦɚ ɧɚɡɵɜɚɟɬɫɹ
ɧɚɡɵɜɚɬɶ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɮɭɧɤɰɢɟɣ
ɦɧɨɝɨɱɥɟɧɨɦ Ɍɟɣɥɨɪɚ ɫɬɟɩɟɧɢ
ɢ ɟɟ ɦɧɨɝɨɱɥɟɧɨɦ Ɍɟɣɥɨɪɚ ɫɬɟɩɟɧɢ n
()
xf
ɨɫɬɚɬɨɱɧɵɦ ɱɥɟɧɨɦ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɢ ɨɛɨɡɧɚɱɚɬɶ
()
n
:
xR
n
. Ȼɭɞɟɦ
() () ()
−=
. (6.5)
xSxfxR
nn
Ɍɟɨɪɟɦɚ
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɛɟɫɤɨɧɟɱɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚɹ ɜ ɬɨɱɤɟ
6.1.
ɮɭɧɤɰɢɹ
x
0
()
xf
ɹɜɥɹɥɚɫɶ ɫɭɦɦɨɣ ɫɨɫɬɚɜɥɟɧɧɨɝɨ ɞɥɹ ɧɟɟ ɪɹɞɚ Ɍɟɣɥɨɪɚ, ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ,
ɱɬɨɛɵ ɨɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɪɹɞɚ
ɫɬɪɟɦɢɥɫɹ ɤ ɧɭɥɸ ɩɪɢ
()
xR
n
∞→n
.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
ɇɟɨɛɯɨɞɢɦɨɫɬɶ.
ɉɭɫɬɶ
ɟɫɬɶ ɫɭɦɦɚ ɪɹɞɚ Ɍɟɣɥɨɪɚ, ɬ. ɟ.
()
xf
Ɍɨɝɞɚ ɢɡ ɮɨɪɦɭɥɵ (6.5) ɫɥɟɞɭɟɬ, ɱɬɨ
n
n
lim
()
n
∞→
() ()
=
n
.
0lim =∞→xR
.
xfxS
89
Ⱦɨɫɬɚɬɨɱɧɨɫɬɶ.
ɉɭɫɬɶ
n
Ɍɨɝɞɚ ɢɡ ɮɨɪɦɭɥɵ (6.5) ɫɥɟɞɭɟɬ, ɱɬɨ
() ()
n
lim
∞→
=
n
()
n
. Ɂɧɚɱɢɬ,
xfxS
.
0lim =∞→xR
[]
ɢ ɟɫɬɶ ɫɭɦɦɚ ɪɹɞɚ.
()
xf
() ()
n
n
, ɬ. ɟ. ɱɬɨ
0lim =−∞→xSxf
Ɂɚɦɟɱɚɧɢɟ.
ɇɟ ɫɥɟɞɭɟɬ ɩɭɬɚɬɶ ɨɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɫ ɨɫɬɚɬɤɨɦ ɪɹɞɚ Ɍɟɣɥɨɪɚ.
Ɉɫɬɚɬɨɤ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɟɫɬɶ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɟɝɨ ɫɭɦɦɨɣ
:
()
xS
n
ɮɭɧɤɰɢɟɣ
() ()
, ɚ ɨɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɪɹɞɚ Ɍɟɣɥɨɪɚ
xSxSn−
, ɞɥɹ ɤɨɬɨɪɨɣ ɷɬɨɬ ɪɹɞ ɫɨɫɬɚɜɥɟɧ, ɢ
()
xf
ɢ ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɨɣ
()
xS
ɟɫɬɶ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ
()
xR
n
. Ɉɫɬɚɬɨɤ ɪɹɞɚ Ɍɟɣɥɨɪɚ
()
xS
n
ɛɭɞɟɬ ɫɨɜɩɚɞɚɬɶ ɫ ɨɫɬɚɬɨɱɧɵɦ ɱɥɟɧɨɦ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ, ɟɫɥɢ
() ()
.
xfxS =
Ɍɟɨɪɟɦɚ (6.1) ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɜɨɩɪɨɫɚ ɨ ɪɚɡɥɨɠɢɦɨɫɬɢ
ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɧɭɠɧɨ ɢɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɟɝɨ ɨɫɬɚɬɨɱɧɨɝɨ ɱɥɟɧɚ
ɩɪɢ
()
xR
n
ɪɹɞɚ Ɍɟɣɥɨɪɚ ɪɚɜɧɚ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ ɜ ɬɨɱɤɟ
ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ, ɬɨ ɪɹɞ Ɍɟɣɥɨɪɚ ɥɢɛɨ ɪɚɫɯɨɞɢɬɫɹ, ɥɢɛɨ ɟɝɨ ɫɭɦɦɚ ɩɪɢ
ɧɟ ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɱɟɧɢɟɦ ɮɭɧɤɰɢɢ ɜ ɞɚɧɧɨɣ ɬɨɱɤɟ
Ɍɟɣɥɨɪɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɪɚɡɧɵɯ ɮɨɪɦɚɯ. ɇɚɩɪɢɦɟɪ,
ɝɪɚɧɠɚ:
ɝɞɟ
()
∈
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɩɪɢ
∞→n
. ȿɫɥɢ ɞɥɹ ɞɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ
x
0
xx =
0
, ɬ. ɟ.
x
0
n
()
0lim =∞→xR
n
. ȿɫɥɢ
()
xf
0
. Ɉɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɪɹɞɚ
, ɬɨ ɫɭɦɦɚ
ɧɟ
()
0xRn
xx =
ɜ ɮɨɪɦɟ Ʌɚ-
()
1
+
n
()
()
=
xR , (6.6)
n
.
xxc ;
0
cf
()
()
+
n
, ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɨɫɬɚɬɨɱɧɨɝɨ ɱɥɟɧɚ ɞɥɹ
00=x
−⋅
!1
1
+
n
xx
0
ɪɹɞɚ Ɇɚɤɥɨɪɟɧɚ
ɝɞɟ
()
xc ;0∈
n
.
()
1
+
n
()
cf
1
+
()
=
xR , (6.7)
()
+
n
n
⋅
x
!1
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɚɜɟɧɫɬɜɚɦɢ (6.4–6.6) ɢɦɟɟɦ:
′′′
fx fx
ɝɞɟ
fx fx x x x x
.
()
xxc ;
∈
0
()
=+ ⋅−+ ⋅−+
()
()
00 0
n
()
fx
... ,
()
+⋅−+ ⋅−
!1!
nn
00
()
1! 2!
0
()
nn
xx xx
00
()
+
n
1
()
fc
()
()
+
()
2
()
...
(6.8)
+
1
0
90
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