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∞
x
21
=++++
......
, (4.3)
aqqq
nn
¦
=
1
n
ɱɬɨ ɚɛɫɨɥɸɬɧɵɟ ɜɟɥɢɱɢɧɵ ɱɥɟɧɨɜ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ (4.1) ɞɥɹ ɥɸɛɨɝɨ ɡɧɚɱɟɧɢɹ
, ɩɪɢɧɚɞɥɟɠɚɳɟɝɨ ɨɬɪɟɡɤɭ
, ɧɟ ɩɪɟɜɨɫɯɨɞɹɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɥɟ-
[]
ba,
ɧɨɜ ɡɧɚɤɨɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ (4.3), ɬ. ɟ.
() ()
nn
...,2,1=≤ naxu
.
ɉɪɢ ɷɬɨɦ ɪɹɞ (4.3) ɧɚɡɵɜɚɟɬɫɹ
ɦɚɠɨɪɢɪɭɸɳɢɦ (ɭɫɢɥɢɜɚɸɳɢɦ) ɩɨ ɨɬɧɨɲɟ-
ɧɢɸ ɤ ɪɹɞɭ (4.1)
ɉɪɢɦɟɪ 4.3.
ɉɨɤɚɡɚɬɶ, ɱɬɨ ɪɹɞ
∞
()
sin
2sin
x
2
3sin
+++
3
sin
nxxx
...
¦
=
1
n
222
n
ɹɜɥɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ.
Ɋɟɲɟɧɢɟ.
Ⱦɥɹ ɥɸɛɨɝɨ ɧɨɦɟɪɚ
ɑɢɫɥɨɜɨɣ ɪɹɞ
()
12 >=α
.
¦
n
n
ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
∞+
1
ɹɜɥɹɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ ɨɛɨɛɳɟɧɧɵɦ ɝɚɪɦɨɧɢɱɟɫɤɢɦ
2
n
=1
()
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɯɨɞɧɵɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ.
1sinnnnx
.
≤
22
X
ɉɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɟɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ ɨɛɥɚɞɚɸɬ ɪɹɞɨɦ ɜɚɠɧɵɯ
ɫɜɨɣɫɬɜ.
1. Ɏɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ (4.1), ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɣɫɹ ɧɚ ɨɬɪɟɡɤɟ
, ɫɯɨ-
[]
ba,
ɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɷɬɨɝɨ ɨɬɪɟɡɤɚ.
2. ȿɫɥɢ ɱɥɟɧɵ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɧɚ ɨɬɪɟɡɤɟ
ɪɹɞɚ (4.1) ɧɟɩɪɟɪɵɜɧɵ, ɬɨ ɫɭɦɦɚ ɪɹɞɚ ɬɚɤɠɟ ɧɟɩɪɟɪɵɜɧɚ ɧɚ ɨɬɪɟɡɤɟ
3. ȿɫɥɢ ɱɥɟɧɵ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɧɚ ɨɬɪɟɡɤɟ
ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ
[]
ba,
ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ
[]
ba,
.
[]
ba,
ɪɹɞɚ (4.1) ɧɟɩɪɟɪɵɜɧɵ ɧɚ ɷɬɨɦ ɨɬɪɟɡɤɟ, ɬɨ ɪɹɞ ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɢɧɬɟɝɪɢɪɨɜɚɬɶ,
ɢ
ɬ. ɟ. ɟɫɥɢ
ɥɸɛɵɟ ɞɜɟ ɬɨɱɤɢ ɨɬɪɟɡɤɚ
x
x
1
2
, ɬɨ
[]
ba,
71
x
ϕ
ϕ
ϕ+⋅ϕ
x
2
() () ()
[]
³
x
1
x
2
() () ()
x
1
21
x
2
21
x
1
......
=+++
n
dxxuxuxu
x
2
n
³³³
x
1
.......
++++=
dxxudxxudxxu
4. ɉɭɫɬɶ ɮɭɧɤɰɢɨɧɚɥɶɧɵɣ ɪɹɞ (4.1) ɫɯɨɞɢɬɫɹ ɧɚ ɨɬɪɟɡɤɟ
ɢɦɟɸɬ ɧɟɩɪɟɪɵɜɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ
() ( )
n
...,2,1=′nxu
. Ɍɨɝɞɚ, ɟɫɥɢ ɪɹɞ, ɩɨɥɭɱɟɧ-
ɢ ɟɝɨ ɱɥɟɧɵ
[]
ba,
ɧɵɣ ɩɨɫɥɟ ɩɨɱɥɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ, ɹɜɥɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ ɧɚ
ɨɬɪɟɡɤɟ
, ɬɨ ɟɝɨ ɫɭɦɦɚ ɪɚɜɧɚ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɫɭɦɦɵ ɞɚɧɧɨɝɨ ɪɹɞɚ:
[]
ba,
[]
() () ()
′
=
++++ xuxuxuxuxuxu
′
′
() () ()
+
2121
′
++
nn
............
+
5. ȿɫɥɢ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɣɫɹ ɧɚ ɨɬɪɟɡɤɟ
ɪɹɞ
[]
ba,
() () ()
21
n
......
++++ xuxuxu
ɭɦɧɨɠɢɬɶ ɧɚ ɨɝɪɚɧɢɱɟɧɧɭɸ ɮɭɧɤɰɢɸ
() () () () () ()
ɛɭɞɟɬ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ ɧɚ ɨɬɪɟɡɤɟ
, ɬɨ ɩɨɥɭɱɟɧɧɵɣ ɪɹɞ
()
x
21
++⋅
.
[]
ba,
......
+⋅
xuxxuxxux
n
ɉɪɢɦɟɪ 4.4.
ɇɚɣɬɢ ɫɭɦɦɭ ɪɹɞɚ
()
32
...642482
+−+−= xxxxS
, ɩɪɢɦɟɧɹɹ ɢɧɬɟɝɪɢɪɨɜɚ-
ɧɢɟ, ɢ ɭɤɚɡɚɬɶ ɨɛɥɚɫɬɶ ɟɝɨ ɫɯɨɞɢɦɨɫɬɢ.
Ɋɟɲɟɧɢɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
()
xF
ɧɭɸ
ɮɭɧɤɰɢɢ
ɇɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ
ɢ ɡɧɚɦɟɧɚɬɟɥɟɦ
xb 21=
1
1
ɝɞɟ
§
x
¨
©
·
, ɬɚɤ ɤɚɤ
;
−∈
¸
2
2
¹
()
xS
:
g 2−=
()
()
()
xF
– ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɩɪɨɝɪɟɫɫɢɹ ɫ ɩɟɪɜɵɦ ɱɥɟɧɨɦ
, ɬɚɤɢɦ ɨɛɪɚɡɨɦ
()
xF
1<g
.
b
1
=
=
g
1
−
+
32
...642482
+−+−= xxxxS
. ɇɚɣɞɟɦ ɩɟɪɜɨɨɛɪɚɡ-
322
...16842
+−+−= xxxxxF
.
x
2
,
x
21
72
()
x
Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦ ɮɭɧɤɰɢɸ
xF
, ɧɚɣɞɟɦ
′
() ()
=xxFxS
=
()
122+
1
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ
()
xS
()
, ɝɞɟ
2
122+=x
§
−∈
x
¨
2
©
ɉɪɢɦɟɪ 4.5.
ɇɚɣɬɢ ɫɭɦɦɭ ɪɹɞɚ
∞+
()
¦
=
n
1
n
1
n
42
+
x
⋅−
n
()
425
+⋅
n
, ɩɪɢɦɟɧɹɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ,
ɭɤɚɡɚɬɶ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ.
Ɋɟɲɟɧɢɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ
() ( )
xS
n
1
⋅−=
n
()
n
425
+⋅
+
−=
⋅
65
+
n
x
Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦ:
¦
n
∞+
=
1
9
75
′
() ()
xS
5
xxx
3
2
55
ɂɦɟɟɦ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ ɫ ɩɟɪɜɵɦ ɱɥɟɧɨɦ
2
ɬɟɥɟɦ
()
ɞɟɬ
xg−
. Ɍɨɝɞɚ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ, ɢɡ ɪɟɲɟɧɢɹ ɧɟɪɚɜɟɧɫɬɜɚ
=
5
5;5−
, ɚ ɫɭɦɦɚ ɪɹɞɚ ɪɚɜɧɚ
5
x
′
()
xS
−
5
=
2
x
1
+
5
5
x
−=
2
x
+
3
5
ɂɧɬɟɝɪɢɪɭɟɦ:
()
25
3
³
5
x
xxxS ++−+−=⋅
−+−=
2
5
+
dx
5
4
()
xS
,
.
2
1
·
;
¸
2
¹
8642
2
⋅
1...
xx
5
24
xx
2
. Ź
10
xxx
−
3
⋅
n
2
n
⋅−=+−+−=
5
x
25
−+−=
2
x
+
25
2
...
+
10585
32
+
.
n
5
x
b −=
1
5
.
5
2
()
Cx
5ln
ɢ ɡɧɚɦɟɧɚ-
1<g
, ɛɭ-
.
73
Ɍɚɤ ɤɚɤ ɫɜɨɛɨɞɧɵɣ ɱɥɟɧ ɪɹɞɚ ɨɬɫɭɬɫɬɜɭɟɬ, ɬɨ
ɧɚɣɞɟɦ:
25
0: =+−= CCC
2
ɂ ɬɚɤ,
ɝɞɟ
∞+
n
()
1
¦
=
n
1
()
5;5−∈x
. Ź
n
+
2
⋅−
n
()
425
+⋅
n
−
=
4
442
5
x
2
, ɢɡ ɷɬɨɝɨ ɭɫɥɨɜɢɹ
()
00 =S
25
.
,5ln
5ln
2
25
()
xx
2
25
22
5ln
++−+
2
,
5ln
74
ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
4.1. Ɉɩɪɟɞɟɥɢɬɶ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɪɹɞɚ:
∞+
16
1
¦
n
¦
n
1
=
1
∞+
=
=++++
¦
n
¦
n
=++++
;
3963
n
xxxx
1
;
41284
n
xxxx
1
∞+
n
xxxx
¦
n
∞+
=
∞+
=
1
¦
n
;
−
1
n
2
=
1
nn
;
⋅=+++
3...2793
xxxx
1
132
n
−
xxxx
;
−
1
n
3
∞+
−
1
n
⋅
23
;
n
xxxx
=
1
∞+
=+++
...
¦
=
1
n
∞+
¦
=
1
n
24
()
⋅=+++
5...125255
xxxx
1
()
+−
nn
⋅⋅
xxxx
2/322/92/72/5
nn
+
.
;
2/1212/72/52/3
=+++
=+++
...
=++++
...
...
1)
2)
3)
4)
5)
6)
7)
8)
111
...
111
...
432
x
1
842
32
2793
241263
x
1
4
432
1
8
4.2. Ɉɩɪɟɞɟɥɢɬɶ ɫɭɦɦɭ ɪɹɞɚ:
∞+
3
x
=+
...
¦
1
1
n
=
∞+
=+
...
¦
1
=
1
n
∞+
=+
...
¦
1
=
1
n
;
n
3
+
x
2
2
x
;
n
2
+
x
2
n
x
;
n
2
+
x
1)
2)
3)
3
x
3
x
+
1
2
2
x
2
+
1
x
2
x
2
+
1
x
3
x
+
+
+
2
3
()() ()
+
x
2
2
x
2
2
()() ()
+
1
x
4
x
2
2
()() ()
+
x
3
x
+
+
+
3
3
+
11
x
2
2
x
3
2
+
1
x
8
x
3
2
+
11
x
75
∞+
()
2/3
xn
−
2
x
¦
()
+
1
1
n
=
.
n
2/3
x
4)
2/1
2
x
+
2/3
+
1
x
2
2
x
+
2
2/3
()()
+
1
x
2/7
2
x
2/3
+
1
x
=+
...
3
4.3. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɪɹɞ ɹɜɥɹɟɬɫɹ ɩɪɚɜɢɥɶɧɨ ɫɯɨɞɹɳɢɦɫɹ:
∞+
()
xn
cos
1)
2)
3)
4)
5)
6)
¦
n
¦
n
¦
n
¦
¦
n
¦
=1
=
=
=
=0
∞+
1
∞+
1
∞+
1
∞+
sin
3
cos
()
;
2
n
()
45arctg2
xn
+
!
n
2
n
, ɧɚ ɩɪɨɦɟɠɭɬɤɟ
n
+
x
−π
x
, ɧɚ ɨɬɪɟɡɤɟ
2
+
3nxnn
()
xn
;
n
2
12
−nn
+
5
2
nx4
, ɧɚ ɨɬɪɟɡɤɟ
−
;
;
[
)
∞+;0
;
[]
π4;0
[]
.
3;7 −−
4.4. ɇɚɣɬɢ ɫɭɦɦɭ ɪɹɞɨɜ, ɩɪɢɦɟɧɹɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ, ɢ ɭɤɚɡɚɬɶ ɨɛɥɚɫɬɶ ɫɯɨ-
ɞɢɦɨɫɬɢ:
1)
3)
5)
7)
¦
ɩ
¦
ɩ
¦
¦
ɩ
∞+
=
∞+
=
∞+
=
∞+
=
()
()
1
113ɩ
3
1
n
⋅+01
xn
+
1
n
1
−
n
; 6)
⋅
xn
−
nn
⋅
xn
; 2)
−
12
n
⋅⋅−
xn
12
; 8)
; 4)
∞+
+
1
n
()
1
¦
=
1
ɩ
∞+
¦
=
1
ɩ
∞+
()()
¦
=
0
ɩ
∞+
()
¦
=
1
ɩ
⋅⋅−
xn
()
⋅+⋅
122
n
⋅+⋅
1
xnn
−
1
n
;
2
nn
;
xn
+
1
n
;
⋅+⋅+
32
xnn
+
2
.
76
4.5. ɇɚɣɬɢ ɫɭɦɦɭ ɪɹɞɨɜ, ɩɪɢɦɟɧɹɹ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ, ɢ ɭɤɚɡɚɬɶ ɨɛɥɚɫɬɶ
ɫɯɨɞɢɦɨɫɬɢ;
∞+
1)
()
1
¦
=
1
ɩ
∞+
3)
()
¦
=
0
ɩ
∞+
5)
()
1
¦
=
1
ɩ
∞+
7)
()
¦
=
1
ɩ
4
n
x
+
1
n
2
+
n
1
; 2)
⋅−
n
n
; 4)
⋅+−
122
xnn
n
x
1
⋅+
; 6)
⋅−
n
⋅
5
n
+
42
n
; 8)
xn
∞+
x
¦
=
1
ɩ
∞+
()
¦
=
2
ɩ
∞+
()
¦
=
1
ɩ
∞+
()
¦
=
1
ɩ
−
13
n
;
−
13
n
−
1
n
x
+
1
n
1
+
1
n
1
n
⋅−
1
;
⋅−
−
1
n
−
13
n
x
;
⋅−
−
13
n
+
22
n
x
()
+⋅
13
nn
.
77
Ƚɥɚɜɚ 5
x
x
ɋɌȿɉȿɇɇɕȿ ɊəȾɕ
ȼɚɠɧɵɦ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɮɭɧɤɰɢɨɧɚɥɶɧɵɯ ɪɹɞɨɜ ɹɜɥɹɸɬɫɹ ɫɬɟɩɟɧɧɵɟ
ɪɹɞɵ.
ɋɬɟɩɟɧɧɵɦ ɧɚɡɵɜɚɟɬɫɹ ɪɹɞ ɜɢɞɚ
()() ()
2
02010
n
n
......
+−++−+−+
xxaxxaxxaa
0
, (5.1)
ɝɞɟ
ɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɹɞɚ
x
0
ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɢɦɟɟɬ ɜɢɞ
00=x
– ɩɨɫɬɨɹɧɧɵɟ ɱɢɫɥɚ. ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ
aaa ...,,,
n
10
2
210
n
......
+++++
xaxaxaa
n
. (5.2)
Ɋɹɞ (5.1) ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ
ɧɹɦ
.
ɪɹɞɨɦ ɩɨ ɫɬɟɩɟɧɹɦ
()
, ɪɹɞ (5.2) – ɩɨ ɫɬɟɩɟ-
xx −
0
5.1. Ɉɩɪɟɞɟɥɟɧɢɟ, ɨɛɥɚɫɬɶ ɢ ɪɚɞɢɭɫ ɫɯɨɞɢɦɨɫɬɢ
ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ
Ɋɚɫɫɦɨɬɪɟɧɢɟ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ ɧɚɱɧɟɦ ɫ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ
Ɍɟɨɪɟɦɚ 5.1. Ⱥɛɟɥɹ.
ȿɫɥɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɫɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ
()
ɚɛɫɨɥɸɬɧɨ ɜ ɢɧɬɟɪɜɚɥɟ
ɜɢɸ
xx <
.
0
, xx−
, ɬ. ɟ. ɩɪɢ ɜɫɹɤɨɦ x, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɦ ɭɫɥɨ-
00
, ɬɨ ɨɧ ɫɯɨɞɢɬɫɹ ɢ ɩɪɢɬɨɦ
00≠x
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
∞
n
ɉɨ ɭɫɥɨɜɢɸ ɬɟɨɪɟɦɵ ɪɹɞ
ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:
¦
=10n
n
0lim
=
xa
0
n
∞→
n
ɫɯɨɞɢɬɫɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɝɨ ɨɛɳɢɣ ɱɥɟɧ
xa
n
. Ɍɨɝɞɚ, ɜɫɟ ɱɥɟɧɵ ɷɬɨɝɨ ɪɹɞɚ ɨɝɪɚɧɢɱɟɧɵ, ɬ. ɟ.
ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɩɨɫɬɨɹɧɧɨɟ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ Ɇ, ɱɬɨ ɩɪɢ ɥɸɛɨɦ
ɦɟɫɬɨ ɧɟɪɚɜɟɧɫɬɜɨ
n
Mxa
<
. Ɂɚɩɢɲɟɦ ɪɹɞ (5.2) ɜ ɜɢɞɟ:
0
n
2
·
·
§
x
¸
¨
xaa
+
010
¸
¨
x
0
¹
©
§
x
2
¸
¨
xa
+
02
¨
x
0
©
++
¸
¹
n
·
§
x
n
¸
¨
xa
0
¸
¨
x
0
¹
©
ɢ ɫɨɫɬɚɜɢɦ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɷɬɨɝɨ ɪɹɞɚ:
(5.2).
n
......
+
n
ɢɦɟɟɬ
78
2
x
x
x
x
x
x
x
x
xaa
010
x
0
x
2
xa
02
x
0
xa
n
n
x
n
0
x
.
......
+⋅++⋅+⋅+
0
ȼ ɫɢɥɭ ɭɫɬɚɧɨɜɥɟɧɧɨɝɨ ɧɟɪɚɜɟɧɫɬɜɚ ɤɚɠɞɵɣ ɱɥɟɧ ɡɞɟɫɶ ɦɟɧɶɲɟ ɫɨɨɬɜɟɬɫɬɜɭ-
ɸɳɟɝɨ ɱɥɟɧɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɹɞɚ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ
x
:
x
0
2
M
x
x
00
x
MM
x
M
n
x
x
.
......
+++++
0
ȿɫɥɢ
xx <
0
ɢ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ, ɚ, ɡɧɚɱɢɬ, ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɢɬɫɹ ɫɚɦ ɪɹɞ (5.2).
x
, ɬɨ 10<
x
ɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɡɧɚɱɢɬ ɫɯɨɞɢɬɫɹ
Ɂɚɦɟɱɚɧɢɟ.
ɇɟɫɦɨɬɪɹ ɧɚ ɬɨ, ɱɬɨ
n
n
n
xaxa
< , ɦɵ ɧɟ ɦɨɠɟɦ ɫɪɚɡɭ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ
n
0
ɩɪɢɡɧɚɤɨɦ ɫɪɚɜɧɟɧɢɹ, ɩɨɫɤɨɥɶɤɭ ɜ ɭɫɥɨɜɢɢ ɬɟɨɪɟɦɵ ɧɟ ɫɤɚɡɚɧɨ, ɱɬɨ ɪɹɞ ɜ ɫɚɦɨɣ
ɬɨɱɤɟ
ɫɯɨɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ.
x
0
ɋɥɟɞɫɬɜɢɟ.
ȿɫɥɢ ɫɬɟɩɟɧɧɨɣ ɪɹɞ (5.2) ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ
ɤɨɦ
, ɛɨɥɶɲɟɦ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ, ɱɟɦ
ȼ ɫɚɦɨɦ ɞɟɥɟ, ɟɫɥɢ ɛɵ ɨɧ ɫɯɨɞɢɥɫɹ ɩɪɢ ɤɚɤɨɦ-ɧɢɛɭɞɶ ɬɚɤɨɦ
, ɬɨ ɨɧ ɪɚɫɯɨɞɢɬɫɹ ɢ ɩɪɢ ɜɫɹ-
xx =
0
, ɬ. ɟ. ɩɪɢ
x
0
xx >
.
0
, ɬɨ ɜ ɫɢɥɭ
ɬɟɨɪɟɦɵ Ⱥɛɟɥɹ ɨɧ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɢɥɫɹ ɛɵ ɢ ɩɪɢ ɜɫɟɯ ɦɟɧɶɲɢɯ ɩɨ ɚɛɫɨɥɸɬɧɨɣ
ɜɟɥɢɱɢɧɟ ɡɧɚɱɟɧɢɹɯ
, ɜ ɱɚɫɬɧɨɫɬɢ ɩɪɢ
, ɱɬɨ ɩɪɨɬɢɜɨɪɟɱɢɬ ɭɫɥɨɜɢɸ.
xx =
0
Ɂɞɟɫɶ ɜɨɡɦɨɠɧɵ ɬɪɢ ɜɢɞɚ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ.
1. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɨɫɬɨɢɬ ɬɨɥɶɤɨ ɢɡ ɨɞɧɨɣ ɬɨɱɤɢ
ɞɢɬɫɹ ɞɥɹ ɜɫɟɯ ɡɧɚɱɟɧɢɣ
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɪɹɞɚ
ɧɚɱɢɧɚɹ ɫ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɝɨ
, ɤɪɨɦɟ ɨɞɧɨɝɨ.
22
n
, ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
xnxx
ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ 0≠x,
......21
+++++
0=
, ɬ. ɟ. ɪɹɞ ɪɚɫɯɨ-
1>nx
, ɨɬ-
ɤɭɞɚ ɜɵɬɟɤɚɟɬ ɧɟɪɚɜɟɧɫɬɜɨ
ɦɢɬɫɹ ɤ ɧɭɥɸ, ɬ. ɟ. ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ ɥɸɛɨɦ
2. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɨɫɬɨɢɬ ɢɡ ɜɫɟɯ ɬɨɱɟɤ ɨɫɢ
ɫɯɨɞɢɬɫɹ ɩɪɢ ɜɫɟɯ
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɪɹɞɚ
ɧɚɹ ɫ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɝɨ
.
1
n
1>nnxn
ɨɡɧɚɱɚɸɳɟɟ, ɱɬɨ ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɧɟ ɫɬɪɟ-
0≠
.
Ox , ɞɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɪɹɞ
2
x
2
2
n
xx
ɢɦɟɟɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ x, ɧɚɱɢ-
......
+++++
n
n
, ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
x
.
1<
n
79
Ɍɚɤ ɤɚɤ
ɯ
2211
++++
nnnn
x
1
n
+
x
<
n
x
,
2
n
+
x
<
ɢ ɬ. ɞ.
n
ɬɨ, ɧɚɱɢɧɚɹ ɫ ɧɨɦɟɪɚ
ɧɨɜ ɫɯɨɞɹɳɟɝɨɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɹɞɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ ɥɸɛɨɦ
n
, ɱɥɟɧɵ ɪɹɞɚ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɛɭɞɭɬ ɦɟɧɶɲɟ ɱɥɟ-
x
ɪɹɞ ɫɯɨ-
ɞɢɬɫɹ.
3. Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɫɨɫɬɨɢɬ ɛɨɥɶɲɟ ɱɟɦ ɢɡ ɨɞɧɨɣ ɬɨɱɤɢ ɨɫɢ
Ox
, ɩɪɢɱɟɦ
ɟɫɬɶ ɬɨɱɤɢ ɨɫɢ, ɧɟ ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ.
ɇɚɩɪɢɦɟɪ, ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ
2
n
......1
+++++
xxx
,
ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ
x
, ɫɯɨɞɢɬɫɹ ɩɪɢ
1<x
ɢ ɪɚɫɯɨɞɢɬɫɹ ɩɪɢ
1≥x
.
ȼ ɬɪɟɬɶɟɦ ɫɥɭɱɚɟ ɧɚ ɱɢɫɥɨɜɨɣ ɨɫɢ ɧɚɪɹɞɭ ɫ ɬɨɱɤɚɦɢ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɢɦɟ-
ɸɬɫɹ ɢ ɬɨɱɤɢ ɟɝɨ ɪɚɫɯɨɞɢɦɨɫɬɢ.
ɂɡ ɬɟɨɪɟɦɵ Ⱥɛɟɥɹ (ɬɟɨɪɟɦɚ 5.1) ɢ ɟɟ ɫɥɟɞɫɬɜɢɹ ɜɵɬɟɤɚɟɬ, ɱɬɨ ɜɫɟ ɬɨɱɤɢ ɫɯɨɞɢɦɨɫɬɢ ɪɚɫɩɨɥɨɠɟɧɵ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɧɟ ɞɚɥɶɲɟ, ɱɟɦ ɥɸɛɚɹ ɢɡ ɬɨɱɟɤ ɪɚɫɯɨɞɢɦɨɫɬɢ. ɋɨɜɟɪɲɟɧɧɨ ɹɫɧɨ, ɱɬɨ ɬɨɱɤɢ ɫɯɨɞɢɦɨɫɬɢ ɛɭɞɭɬ ɰɟɥɢɤɨɦ ɡɚɩɨɥɧɹɬɶ
ɧɟɤɨɬɨɪɵɣ ɢɧɬɟɪɜɚɥ ɫ ɰɟɧɬɪɨɦ ɜ ɧɚɱɚɥɟ ɤɨɨɪɞɢɧɚɬ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɤɚɠɞɨɝɨ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ, ɢɦɟɸɳɟɝɨ
ɤɚɤ ɬɨɱɤɢ ɫɯɨɞɢɦɨɫɬɢ, ɬɚɤ ɢ ɬɨɱɤɢ ɪɚɫɯɨɞɢɦɨɫɬɢ, ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɩɨɥɨɠɢɬɟɥɶ-
()
ɧɨɟ ɱɢɫɥɨ
R
, ɱɬɨ ɞɥɹ ɜɫɟɯ x, ɩɨ ɦɨɞɭɥɸ ɦɟɧɶɲɢɯ
ɫɯɨɞɢɬɫɹ, ɚ ɞɥɹ ɜɫɟɯ
x
, ɩɨ ɦɨɞɭɥɸ ɛɨɥɶɲɢɯ
()
RxR <
, ɪɹɞ ɚɛɫɨɥɸɬɧɨ
RxR >
, ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ
(ɪɢɫ. 5.1).
ɋɯɨɞɢɬɫɹ Ɋɚɫɯɨɞɢɬɫɹ Ɋɚɫɯɨɞɢɬɫɹ
0
Ɋɢɫ. 5.1
R – R
ɑɬɨ ɤɚɫɚɟɬɫɹ ɡɧɚɱɟɧɢɣ
Rx =
ɢ
Rx −=
, ɬɨ ɡɞɟɫɶ ɦɨɝɭɬ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɪɚɡɥɢɱɧɵɟ ɜɨɡɦɨɠɧɨɫɬɢ: ɪɹɞ ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ ɜ ɨɛɟɢɯ ɬɨɱɤɚɯ, ɢɥɢ ɬɨɥɶɤɨ ɜ ɨɞɧɨɣ
ɢɡ ɧɢɯ, ɢɥɢ ɧɢ ɜ ɨɞɧɨɣ. ɉɪɢ ɷɬɨɦ ɪɹɞ ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ ɤɚɤ ɚɛɫɨɥɸɬɧɨ, ɬɚɤ ɢ
ɭɫɥɨɜɧɨ.
Ɋɚɞɢɭɫɨɦ ɫɯɨɞɢɦɨɫɬɢ
ɞɥɹ ɜɫɟɯ
ɬɟɪɜɚɥ
()
Rxx <,
, ɫɬɟɩɟɧɧɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ, ɚ ɞɥɹ ɜɫɟɯ
ɧɚɡɵɜɚɟɬɫɹ
RR
,−
ɍɫɥɨɜɢɦɫɹ ɞɥɹ ɪɹɞɨɜ, ɪɚɫɯɨɞɹɳɢɯɫɹ ɩɪɢ ɜɫɟɯ
ɚ ɞɥɹ ɪɹɞɨɜ, ɫɯɨɞɹɳɢɯɫɹ ɩɪɢ ɜɫɟɯ
ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ (5.2) ɧɚɡɵɜɚɟɬɫɹ ɬɚɤɨɟ ɱɢɫɥɨ R, ɱɬɨ
Rxx >,
, ɪɚɫɯɨɞɢɬɫɹ. ɂɧ-
ɢɧɬɟɪɜɚɥɨɦ ɫɯɨɞɢɦɨɫɬɢ.
x
, ɫɱɢɬɚɬɶ
x
, ɤɪɨɦɟ
∞=R
.
0=x
, ɫɱɢɬɚɬɶ
0=R
,
80
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