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ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
1.1. Ɋɹɞɵ ɡɚɞɚɧɵ ɮɨɪɦɭɥɨɣ ɨɛɳɟɝɨ ɱɥɟɧɚ
ɬɵɣ ɱɥɟɧɵ:
n
a
1)
n
()
; 2)
!12+=n
. Ɉɩɪɟɞɟɥɢɬɶ ɢɯ ɬɪɟɬɢɣ ɢ ɞɟɫɹ-
a
n
()
!2
n
=
a
n
;
−
12
n
a
3)
n
5)
=
a
n
()
a
7)
n
⋅
n
n
n
()
⋅−=
; 4)
121+
n
12
−
n
; 6)
2
14
+
n
!!12
−
n
; 8)
1
+
n
()
2
a
n
a
=
n
a
=
n
()
1
n
!2n
2
n
1
+
n
⋅−=
;
23
+
n
;
,
!!21n
ɝɞɟ
() ()
.
nnnn 2...642!!2,12...71!!12 ⋅⋅⋅⋅=−⋅⋅⋅=−
1.2. Ɋɹɞɵ ɡɚɞɚɧɵ ɪɟɤɭɪɪɟɧɬɧɵɦ ɫɨɨɬɧɨɲɟɧɢɟɦ. Ɂɚɩɢɫɚɬɶ ɮɨɪɦɭɥɭ ɢɯ ɨɛ-
ɳɟɝɨ ɱɥɟɧɚ:
1)
3)
5)
7)
11
+ nn
+
11
,
11=a
1 nn
+
,
11=a
; 2)
2,3
−==
aaa
; 4)
aaa ⋅−==
3,2
nn
; 6)
!
aa =
aaaa
nn
; 8)
121
...−+++=
11=a
11=a
;
3,5
+=−=
aaa
11
+ nn
;
aaa ⋅=−=
2,3
+
,
1
,
nn
11
()
12
aa
1+=−nn
;
ana ⋅+=+1
nn
.
1.3. Ⱦɥɹ ɞɚɧɧɨɝɨ ɪɹɞɚ ɡɚɩɢɫɚɬɶ ɮɨɪɦɭɥɵ ɨɛɳɟɝɨ ɱɥɟɧɚ:
6
4
2
1)
3
2
3)
5
5)
1 +
+
7)
+++
...
; 2)
7
5
16
8
4
11
8
27
1
21 ++++
9
4
1
+
⋅
++−+−
14
1
3
16
+
⋅⋅
21 +++++
;...
8)
1
3
...
; 4)
6
3
; 6)
25
⋅⋅⋅
43211321121
3
2
5
3
7
5
5
4
6
7
9
...
;
24
...
+−+−
;
....1;11;1;1 −−−
;...7302428226102 ++++++
21
1.4. Ⱦɥɹ ɞɚɧɧɵɯ ɪɹɞɨɜ ɡɚɩɢɫɚɬɶ ɮɨɪɦɭɥɭ
n
-ɣ ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɵ. ɂɫɫɥɟɞɨ-
ɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɪɹɞɨɜ ɢ ɜ ɫɥɭɱɚɟ ɫɯɨɞɢɦɨɫɬɢ ɨɩɪɟɞɟɥɢɬɶ ɢɯ ɫɭɦɦɭ:
1)
3)
5)
7)
9)
11)
¦
n
¦
n
¦
n
¦
n
¦
n
=
¦
n
∞+
=
∞+
=
∞+
=
∞+
=
∞+
2
∞+
=
nn
52
+
; 2)
n
10
1
nn
25
−
n
10
1
1
2
1
1
()()
1
()
ln
1
−
n
2
+
n
−
n
nn
()
23
13
; 4)
; 6)
++
35244
nn
; 8)
+⋅+
32
24
; 10)
−⋅−
21
nn
; 12)
¦
n
¦
n
¦
n
¦
n
¦
n
¦
n
∞+
=
∞+
=
∞+
1
=
∞+
=
∞+
=
∞+
=
nn
72
+
;
n
14
1
nn
27
−
;
n
14
1
1
2
++
nn
1
()()
1
()
2
ln
1
+⋅+
nn
+
5
n
2
()
nn
−
35
n
.
+
25
n
;
63324
;
54
;
−⋅+
12
1.5. Ⱦɥɹ ɞɚɧɧɵɯ ɪɹɞɨɜ ɩɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨɝɨ ɩɪɢɡɧɚɤɚ ɫɯɨ-
ɞɢɦɨɫɬɢ:
1)
3)
5)
7)
¦
n
¦
n
¦
n
¦
n
∞+
+
1
n
2
1
=
∞+
⋅
n
1
=
∞+
n
§
¨
n
©
=
1
∞+
+
n
§
¨
−
n
©
=
1
; 2)
−+
43
nn
1
; 4)
sin
n
n
−
−
43
·
; 6)
¸
+
75
¹
15
−
n
3
·
; 8)
¸
2
¹
¦
n
¦
n
¦
n
¦
n
∞+
=
∞+
=
∞+
=
∞+
=
+
23
n
2
1
3cos
n
n
1
−
n
§
¨
+
n
23
©
1
+
n
§
¨
−
n
©
1
;
+−
53
nn
;
n
12
·
;
¸
¹
n
32
−
12
·
.
¸
32
¹
22
Ƚɥɚɜɚ 2
M
ɂɋɋɅȿȾɈȼȺɇɂȿ
ɋɏɈȾɂɆɈɋɌɂ ɑɂɋɅɈȼɕɏ ɊəȾɈȼ
ɋ ɉɈɅɈɀɂɌȿɅɖɇɕɆɂ ɑɅȿɇȺɆɂ
Ɉɩɪɟɞɟɥɟɧɢɟ ɢ ɧɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɧɟ ɩɨɡɜɨɥɹɸɬ ɭɫɬɚɧɨɜɢɬɶ ɩɨɜɟɞɟɧɢɟ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ. ɇɚ ɩɪɚɤɬɢɤɟ ɞɥɹ ɪɟɲɟɧɢɹ ɜɨɩɪɨɫɚ
ɨ ɬɨɦ, ɫɯɨɞɢɬɫɹ ɞɚɧɧɵɣ ɪɹɞ ɢɥɢ ɪɚɫɯɨɞɢɬɫɹ, ɱɚɳɟ ɜɫɟɝɨ ɢɫɩɨɥɶɡɭɸɬ ɞɨɫɬɚɬɨɱɧɵɟ
ɩɪɢɡɧɚɤɢ.
ȼ ɷɬɨɣ ɝɥɚɜɟ ɛɭɞɟɦ ɢɫɫɥɟɞɨɜɚɬɶ ɱɢɫɥɨɜɵɟ ɪɹɞɵ ɩɨɫɬɨɹɧɧɨɝɨ ɡɧɚɤɚ, ɬɨɱɧɟɟ,
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɪɹɞɵ, ɞɥɹ ɤɨɬɨɪɵɯ
. Ɋɚɫɫɦɨɬɪɢɦ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ, ɚ
0≥na
ɡɚɬɟɦ ɞɨɫɬɚɬɨɱɧɵɟ ɩɪɢɡɧɚɤɢ ɫɯɨɞɢɦɨɫɬɢ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ.
Ɂɚɦɟɱɚɧɢɟ.
ɇɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɨɬɞɟɥɶɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɪɹɞɵ, ɞɥɹ ɤɨɬɨɪɵɯ
ɠɟɧɢɢ ɟɝɨ ɧɚ
, ɬɚɤ ɤɚɤ, ɨɬɪɢɰɚɬɟɥɶɧɵɣ ɪɹɞ ɫɬɚɧɨɜɢɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɩɪɢ ɭɦɧɨ-
0≤na
()
1−
, ɩɪɢ ɷɬɨɦ, ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ 1.3, ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ ɧɟ ɦɟɧɹɟɬɫɹ.
2.1. ɍɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ
Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɱɢɫɥɨɜɨɣ ɪɹɞ, ɱɥɟɧɵ ɤɨɬɨɪɨɝɨ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵ:
Ɍɟɨɪɟɦɚ 2.1.
21
Ɋɹɞ (2.1) ɢɦɟɟɬ ɤɨɧɟɱɧɭɸ ɫɭɦɦɭ (ɬ. ɟ. ɫɯɨɞɢɬɫɹ), ɟɫɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɟɝɨ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ
ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɪɹɞ ɢɦɟɟɬ ɛɟɫ-
{}
S
n
ɤɨɧɟɱɧɭɸ ɫɭɦɦɭ (ɬ. ɟ. ɪɚɫɯɨɞɢɬɫɹ).
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
Ⱦɥɹ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɪɹɞɚ (2.1) ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ
ɡɧɚɱɢɬ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɹɜɥɹɟɬɫɹ ɦɨɧɨɬɨɧɧɨ ɜɨɡɪɚɫɬɚɸɳɟɣ.
ȿɫɥɢ ɩɪɢ ɷɬɨɦ ɧɚɣɞɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɱɢɫɥɨ
n
ɦɟɪɚ
ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
ɧɵɯ ɫɭɦɦ ɹɜɥɹɟɬɫɹ ɬɚɤɠɟ ɨɝɪɚɧɢɱɟɧɧɨɣ ɫɜɟɪɯɭ. ɂɡɜɟɫɬɧɨ, ɱɬɨ ɥɸɛɚɹ ɦɨɧɨɬɨɧɧɚɹ
ɢ ɨɝɪɚɧɢɱɟɧɧɚɹ ɱɢɫɥɨɜɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɢɦɟɟɬ ɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ (ɬɟɨɪɟɦɚ
∞
,
=++++
......
¦
n
()
aaaa
nn
1
=
SaSS ≥+=
nnnn
++ 11
ɬɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱ-
,MSn≤
. (2.1)
0≥na
,
0>
, ɬɚɤɨɟ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɧɨ-
23
SS
ȼɟɣɟɪɲɬɪɚɫɫɚ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɩɨɥɧɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨ
lim
n
ɫɬɜɭɸɳɟɟ ɨɩɪɟɞɟɥɟɧɢɸ ɫɯɨɞɢɦɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɹɞ (2.1) ɫɯɨɞɢɬɫɹ.
=
, ɫɨɨɬɜɟɬ-
n
∞→
ɂɬɚɤ, ɭɫɥɨɜɢɟɦ ɫɯɨɞɢɦɨɫɬɢ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɹɜɥɹɟɬɫɹ ɨɝɪɚɧɢɱɟɧɧɨɫɬɶ ɫɜɟɪɯɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɟɝɨ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ
.
{}
S
n
Ɉɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ.
Ɋɚɫɫɦɨɬɪɢɦ ɱɢɫɥɨɜɨɣ ɪɹɞ ɜɢɞɚ
∞
1
3
2
5
4
1
1
1
1
1
...
...
1
()
¦
αααααα
nn
n
1
=
,
R
∈α=+++++++
ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɟɬɫɹ
ɨɛɨɛɳɟɧɧɵɦ ɝɚɪɦɨɧɢɱɟɫɤɢɦ.
Ⱦɥɹ ɨɛɨɛɳɟɧɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɪɹɞɚ ɧɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ
1
ɜɵɩɨɥɧɹɟɬɫɹ
§
lim
¨
n
∞→
©
·
, ɪɹɞ ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ, ɚ ɦɨɠɟɬ ɪɚɫɯɨɞɢɬɶɫɹ. ɂɫɫɥɟ-
0
=
¸
α
n
¹
ɞɭɟɦ ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ.
ɉɪɢ
1=α ɢɦɟɟɦ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ, ɤɨɬɨɪɵɣ, ɤɚɤ ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɪɚɧɟɟ,
ɪɚɫɯɨɞɢɬɫɹ.
11
ɉɪɢ
1<α ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
<
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ
α
n
n
ɨɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ, ɬɚɤ ɤɚɤ ɤɚɠɞɵɣ ɟɝɨ ɱɥɟɧ ɛɨɥɶɲɟ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɱɥɟɧɚ ɪɚɫɯɨɞɹɳɟɝɨɫɹ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɪɹɞɚ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɝɪɚɧɢɱɟɧɧɨɣ, ɭɫɥɨɜɢɟ ɬɟɨɪɟɦɵ 2.1 ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ.
ɉɭɫɬɶ
1>α . ɋɝɪɭɩɩɢɪɭɟɦ ɱɥɟɧɵ ɪɹɞɚ, ɧɚɱɢɧɚɹ ɫɨ ɜɬɨɪɨɝɨ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ
ɩɨ 2, 4, 8 ɢ ɬ. ɞ. ɱɥɟɧɨɜ:
1 +
++
¨
3
2
©
1
1
§
1
§
·
¨
¸
¹
5
4
©
++++
7
6
1
1
1
1
§
·
¨
¸
8
©
¹
1
·
.
15
...
¸
αααααααα
¹
...
+++
ȿɫɥɢ ɜ ɤɚɠɞɨɣ ɫɤɨɛɤɟ ɡɚɦɟɧɢɬɶ ɜɫɟ ɫɥɚɝɚɟɦɵɟ ɧɚ ɩɟɪɜɨɟ ɜ ɷɬɨɣ ɫɤɨɛɤɟ, ɬɨ
ɩɨɥɭɱɢɦ:
1
2
2
==+
2
;
1
−αααα
2
1
2
1
===+++
2
1
;
()
121
−α−αααααα
1
4
1
1
1
1
4
4
4
4
4
4
24
===+++++++
k
2
1
,
()
131
−α−αααααααααα
1
8
1
1
1
1
1
1
1
1
8
8
8
8
8
8
8
8
8
8
… .
Ɉɛɨɡɧɚɱɢɦ
1−α=
. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
1
2
3
2
=+<+
;
k
αααα
2
2
1
1
1
1
1
4
4
4
7
6
5
4
=+++<+++
;
k2
αααααααα
2
4
1
1
1
1
1
1
1
1
1
8
1
...
8
8
8
8
8
8
8
15
=+++++++<++
;
k3
αααααααααα
2
8
1
1
1
1
1
1
1
1
1
… .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɝɪɭɩɩɢɪɨɜɚɧɧɵɟ ɱɥɟɧɵ ɨɛɨɛɳɟɧɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɪɹɞɚ
ɦɟɧɶɲɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɥɟɧɨɜ ɫɯɨɞɹɳɟɝɨɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɪɹɞɚ, ɫɭɦɦɚ ɤɨɬɨɪɨɝɨ
1
S
=
.
1
1
−
k
2
Ɂɧɚɱɢɬ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɨɛɨɛɳɟɧɧɨɝɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ
ɪɹɞɚ ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ, ɩɨ ɬɟɨɪɟɦɟ 2.1 ɪɹɞ ɫɯɨɞɢɬɫɹ.
Ɉɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ ɛɭɞɟɦ ɜ ɞɚɥɶɧɟɣɲɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɚɤ ɷɬɚɥɨɧɧɵɣ.
Ɋɚɫɫɦɨɬɪɟɧɧɨɟ ɜɵɲɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɨɣ
ɞɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ
ɞɨɫɬɚɬɨɱɧɵɯ ɩɪɢɡɧɚɤɨɜ ɫɯɨɞɢɦɨɫɬɢ.
2.2. ɉɪɢɡɧɚɤɢ ɫɪɚɜɧɟɧɢɹ
Ɍɟɨɪɟɦɚ 2.2.
ɉɟɪɜɵɣ ɩɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ.
ɧɵɦɢ ɱɥɟɧɚɦɢ
ɉɭɫɬɶ ɞɚɧɵ ɞɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɚ ɫ ɩɨɥɨɠɢɬɟɥɶ-
∞
¦
()
a
n
1*n
=
ɢ
∞
¦
1**n
=
()
b
n
,
25
ɩɪɢɱɟɦ ɞɥɹ ɥɸɛɨɝɨ ɧɨɦɟɪɚ
S
()
*
ɱɥɟɧ ɪɹɞɚ
ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ
1) ɢɡ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
2) ɢɡ ɪɚɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
n
ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
()
**
ɫɥɟɞɭɟɬ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ
()
*
ɫɥɟɞɭɟɬ ɪɚɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ
()
()
*
;
()
, ɬ. ɟ. ɤɚɠɞɵɣ
ba ≤
nn
**
. Ɍɨɝɞɚ:
**
.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
()
ȿɫɥɢ
*
ɫɭɦɦɵ ɪɹɞɨɜ ɢ ɪɹɞ
() ( ) ( )
ɱɟɦ
()
*
ɪɹɞɚ
ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ ɱɢɫɥɨɦ
≤≤
nn
ɞɢɬɫɹ. ȿɫɥɢ ɠɟ ɪɹɞ
ɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
aaaaS ++++= ...
321
()
**
ɫɯɨɞɢɬɫɹ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ
*****
SSS
, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ
()
*
ɪɚɫɯɨɞɢɬɫɹ, ɧɨ ɩɨɫɤɨɥɶɤɭ, ɩɨ ɭɫɥɨɜɢɸ ɬɟɨɪɟɦɵ, ɜɵɩɨɥɧɹ-
() ( )
***
SS ≤
, ɬɨ ɬɚɤ ɠɟ ɪɚɫɯɨɞɢɬɫɹ ɪɹɞ
nn
()
**
ɢ
nn
()
**
, ɚ ɡɧɚɱɢɬ, ɩɨ ɬɟɨɪɟɦɟ 2.1 ɪɹɞ
321
n
()
**
bbbbS ++++= ...
() ()
lim SS
n
∞→
.
– ɱɚɫɬɢɱɧɵɟ
nn
=
ɉɪɢɦɟɪ 2.1.
ɂɫɫɥɟɞɨɜɚɬɶ ɪɹɞɵ ɧɚ ɫɯɨɞɢɦɨɫɬɶ:
∞
3
+
8
1)
¦
n
=
1
n
()
n
;
3
n
⋅+
32
∞+
1
2)
¦
n
3
2
1
=
.
−−
1
nn
Ɋɟɲɟɧɢɟ.
1. ɉɨɞɛɟɪɟɦ ɷɬɚɥɨɧɧɵɣ ɪɹɞ, ɞɥɹ ɷɬɨɝɨ ɩɪɟɨɛɪɚɡɭɟɦ ɮɨɪɦɭɥɭ ɟɝɨ ɨɛɳɟɝɨ
ɱɥɟɧɚ:
3
n
8
+
23
nnn
,
n
38126
⋅+++
a
=
n
()
n
3
n
8
+
=
3
n
()
32
⋅+
3
ɨɱɟɜɢɞɧɨ, ɱɬɨ
a
n
<
n
3
()
n
Ɍɚɤ ɤɚɤ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɪɹɞ
+
1
8
38
⋅+
.
=
nn
3
∞
1
ɹɜɥɹɟɬɫɹ ɫɯɨɞɹɳɢɦɫɹ, ɬɨ ɩɨ ɩɟɪɜɨɦɭ
¦
n
3
1
n
=
ɭɫɥɨɜɢɸ ɩɪɢɡɧɚɤɚ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.2) ɢɫɯɨɞɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ.
Ɉɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɧɟɪɚɜɟɧɫɬɜɭ
2.
****
()
*
, ɩɪɢ-
ɫɯɨ-
3
3
2
2
.
111−−<nnn
26
∞+
1
ȼ ɤɚɱɟɫɬɜɟ ɷɬɚɥɨɧɧɨɝɨ ɜɨɡɶɦɟɦ ɪɹɞ
ɳɢɦɫɹ ɨɛɨɛɳɟɧɧɵɦ ɝɚɪɦɨɧɢɱɟɫɤɢɦ
¦
n
§
¨
©
ɩɪɢɡɧɚɤɚ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.2) ɢɫɯɨɞɧɵɣ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ.
, ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ ɪɚɫɯɨɞɹ-
3
2
n
1
=
2
·
. Ɍɨɝɞɚ ɩɨ ɜɬɨɪɨɦɭ ɭɫɥɨɜɢɸ
<=α 1
¸
3
¹
X
Ɍɟɨɪɟɦɚ 2.3.
ȼɬɨɪɨɣ ɩɪɢɡɧɚɤ ɫɪɚɜɧɟɧɢɹ (ɩɪɟɞɟɥɶɧɵɣ).
ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ
∞
¦
1n
=
∞
ɢ
a
n
¦
b
n
1n
=
ɧɵɣ ɨɬ ɧɭɥɹ, ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɢɯ ɨɛɳɢɯ ɱɥɟɧɨɜ
ɉɭɫɬɶ ɞɚɧɵ ɞɜɚ ɱɢɫɥɨɜɵɯ ɪɹɞɚ
. ȿɫɥɢ ɫɭɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ, ɨɬɥɢɱ-
a
n
b
n
∞→
n
()
AA
<=
0lim
ɫɯɨɞɹɬɫɹ ɢɥɢ ɪɚɫɯɨɞɹɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɩɪɟɞɟɥɚ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ,
ɞɥɹ ɥɸɛɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ
(ɬ. ɟ. ɞɥɹ ɜɫɟɯ
a
n
b
n
ȿɫɥɢ ɫɯɨɞɢɬɫɹ ɪɹɞ
) ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ:
nn ≥
0
ε<− A
ɢɥɢ
A
∞
b
n
¦
1n
=
ɩɪɢɡɧɚɤɭ ɫɯɨɞɢɦɨɫɬɢ (ɬɟɨɪɟɦɚ 2.2) ɫɯɨɞɢɬɫɹ ɪɹɞ
∞
ɬɨ ɩɨ ɩɪɢɡɧɚɤɭ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.2) ɫɯɨɞɢɬɫɹ ɪɹɞ
,
a
n
¦
1
n
=
ɩɨ ɬɟɨɪɟɦɟ 1.3 ɬɚɤɠɟ ɫɯɨɞɢɬɫɹ ɪɹɞ
ɞɥɹ ɪɚɫɯɨɞɹɳɢɯɫɹ ɪɹɞɨɜ.
ε
ɧɚɣɞɟɬɫɹ ɧɨɦɟɪ
a
n
b
n
,
ε+<<ε− A
() ()
, ɧɚɱɢɧɚɹ ɫ ɤɨɬɨɪɨɝɨ
n
0
, ɬɨ ɩɨ ɬɟɨɪɟɦɟ 1.3 ɫɯɨɞɢɬɫɹ ɪɹɞ
∞
; ɟɫɥɢ ɠɟ ɫɯɨɞɢɬɫɹ ɪɹɞ
a
n
¦
1n
=
∞
. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɞɨɤɚɡɚɬɶ ɬɟɨɪɟɦɭ
b
n
¦
1n
=
.
bAabA ε+<<ε−
nnn
∞
()
¦
1n
=
∞
()
ε−
¦
=
bA
1n
ε+
ɉɪɢɦɟɪ 2.2.
ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɨɜ:
∞
π
1)
2)
¦
n
¦
n
=
∞
=
n
sin
1
1
;
2
+
1
n
2
+
1
n
.
24
+−
523
nn
, ɬɨ ɪɹɞɵ
, ɢ ɩɨ
bA
n
, ɚ ɬɨɝɞɚ
n
27
Ɋɟɲɟɧɢɟ.
∞
π
1. ɂɫɫɥɟɞɭɟɦ ɩɨɜɟɞɟɧɢɟ ɪɹɞɚ
¦
n
n
.
2
+
1
n
1
=
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɟɞɟɥɶɧɵɦ ɩɪɢɡɧɚɤɨɦ (ɬɟɨɪɟɦɚ 2.3) ɢ ɫɪɚɜɧɢɦ ɪɹɞ ɫ ɷɬɚɥɨɧɧɵɦ
∞
ɪɚɫɯɨɞɹɳɢɦɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɦ ɪɹɞɨɦ
¦
n
11n
=
. Ɍɚɤ ɤɚɤ
lim
1
n
π
§
¨
2
n
©
·
:
1
+
⋅π=
¸
n
¹
lim
2
n
2
n
⋅π=
1
+
nnn
lim
1
∞→∞→∞→
+
1
,
π=
1
2
n
ɬɨ ɩɨ ɩɪɟɞɟɥɶɧɨɦɭ ɩɪɢɡɧɚɤɭ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.3) ɨɛɚ ɪɹɞɚ ɪɚɫɯɨɞɹɬɫɹ.
∞
π
n
Ɍɟɩɟɪɶ ɢɫɩɨɥɶɡɭɟɦ ɪɹɞ
¦
n
=
1
ɤɚɤ ɷɬɚɥɨɧɧɵɣ ɞɥɹ ɢɫɯɨɞɧɨɝɨ ɪɹɞɚ. ȼɵ-
2
+
1
n
ɱɢɫɥɢɦ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ ɢɯ ɨɛɳɢɯ ɱɥɟɧɨɜ:
π
lim
sin
n
2
+
n
π
n
2
1
+
n
π
n
,
=
1
=
n
m
m
2
1
+
0
→
lim
nn
m
sin
1
==
.
m
∞→∞→
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨ ɩɪɟɞɟɥɶɧɨɦɭ ɩɪɢɡɧɚɤɭ ɫɪɚɜɧɟɧɢɹ ( ɬɟɨɪɟɦɚ 2.3) ɢɫɯɨɞɧɵɣ
ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ.
2. ȿɫɥɢ ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɩɪɟɞɫɬɚɜɥɟɧ ɞɪɨɛɧɨ ɪɚɰɢɨɧɚɥɶɧɵɦ ɜɵɪɚɠɟɧɢɟɦ,
ɬɨ ɜ ɤɚɱɟɫɬɜɟ ɷɬɚɥɨɧɧɨɝɨ ɪɹɞɚ ɫɥɟɞɭɟɬ ɜɡɹɬɶ ɨɛɨɛɳɟɧɧɵɣ ɝɚɪɦɨɧɢɱɟɫɤɢɣ ɪɹɞ ɫ
α
ɩɨɤɚɡɚɬɟɥɟɦ ɫɬɟɩɟɧɢ
ɱɢɫɥɢɬɟɥɹ. Ⱦɥɹ ɞɚɧɧɨɝɨ ɪɹɞɚ
∞
¦
n
=
1
. Ɍɚɤ ɤɚɤ
2
n
1
ɪɹɞ
ɪɚɜɧɵɦ ɪɚɡɧɨɫɬɢ ɫɬɟɩɟɧɟɣ ɦɧɨɝɨɱɥɟɧɨɜ ɡɧɚɦɟɧɚɬɟɥɹ ɢ
1224 >=−=α , ɬ. ɟ. ɷɬɚɥɨɧɧɵɦ ɛɭɞɟɬ ɫɯɨɞɹɳɢɣɫɹ
24
nn
+
nn
=
24
523
+−
lim
2
§
¨
¨
©
+
1
n
+−
·
1
¸
:
523
lim
=
224
¸
∞→∞→
nn
nnn
¹
1
§
4
n
¨
lim
=
∞→
n
©
§
4
3
n
¨
©
ɬɨ ɩɨ ɩɪɟɞɟɥɶɧɨɦɭ ɩɪɢɡɧɚɤɭ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.3) ɨɛɚ ɪɹɞɚ ɫɯɨɞɹɬɫɹ.
·
1
+⋅
¸
2
n
++⋅
1
¹
52
42
nn
,
=
3
·
¸
¹
X
28
2.3. Ⱦɨɫɬɚɬɨɱɧɵɟ ɩɪɢɡɧɚɤɢ
A
A
A
A
(
)
Ⱦɚɥɚɦɛɟɪɚ, Ʉɨɲɢ, Ʉɨɲɢ – Ɇɚɤɥɨɪɟɧɚ
Ɍɟɨɪɟɦɚ 2.4.
ɉɪɢɡɧɚɤ Ⱦɚɥɚɦɛɟɪɚ. ɉɭɫɬɶ ɞɥɹ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ (2.1) ɫɭ-
()
1+n
ɳɟɫɬɜɭɟɬ ɤɨɧɟɱɧɵɣ ɢɥɢ ɛɟɫɤɨɧɟɱɧɵɣ ɩɪɟɞɟɥ ɨɬɧɨɲɟɧɢɹ
n
ɩɪɟɞɵɞɭɳɟɦɭ
-ɭ ɱɥɟɧɭ:
a
+
1
n
lim
n
a
∞→
.
A
=
n
Ɍɨɝɞɚ:
1) ɩɪɢ
2) ɩɪɢ 1
3) ɩɪɢ 1
1<
ɪɹɞ (2.1) ɫɯɨɞɢɬɫɹ;
>
ɪɹɞ (2.1) ɪɚɫɯɨɞɢɬɫɹ;
=
ɩɪɢɡɧɚɤ ɧɟ ɞɚɟɬ ɨɬɜɟɬ ɧɚ ɜɨɩɪɨɫ ɨ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ, ɬɪɟɛɭɟɬɫɹ
ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɩɪɟɞɟɥɚ ɱɢɫɥɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ, ɞɥɹ
ε
ɥɸɛɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɢɫɥɚ
ɜɫɟɯ
mn ≥
) ɛɭɞɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ:
ɧɚɣɞɟɬɫɹ ɧɨɦɟɪ m, ɧɚɱɢɧɚɹ ɫ ɤɨɬɨɪɨɝɨ (ɬ. ɟ. ɞɥɹ
a
+
n 1
a
n
ε<−
A
ɢɥɢ
a
+
1
A
n
A
a
n
ɉɭɫɬɶ
1<
, ɩɨɞɛɟɪɟɦ ε ɬɚɤ, ɱɬɨɛɵ
. Ɍɨɝɞɚ
11<=ε+ AA
a
1
m
a
m
a
2
++
m
A
1
a
1
m
+
a
3
+
m
A
1
a
A
<<<
1
2
m
+
ɉɨɥɭɱɚɟɦ
aAaa AaAa
11 21 11
mmm m m
+++
,,
<<⋅<
aAaAa
<⋅ < ⋅
31 21
mm m
++
2
2
,....
ɑɥɟɧɵ ɪɹɞɚ
m
-ɝɨ ɱɥɟɧɚ, ɦɟɧɶɲɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɥɟɧɨɜ ɫɯɨɞɹɳɟɝɨɫɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ
ɫɥɟ
ɪɹɞɚ
m
3
2
11
...
+++ AAAa
1
aaa
.
, ɹɜɥɹɸɳɟɝɨɫɹ ɨɫɬɚɬɤɨɦ ɪɹɞɚ (2.1) ɩɨ-
...
+++
321
+++ mmm
ɉɨ ɩɟɪɜɨɦɭ ɭɫɥɨɜɢɸ ɩɪɢɡɧɚɤɚ ɫɪɚɜɧɟɧɢɹ (ɬɟɨɪɟɦɚ 2.2) ɨɫɬɚɬɨɤ ɪɹɞɚ (2.1)
ɫɯɨɞɢɬɫɹ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨ ɬɟɨɪɟɦɟ 1.1 ɫɯɨɞɢɬɫɹ ɢ ɫɚɦ ɪɹɞ (2.1).
-ɝɨ ɱɥɟɧɚ ɪɹɞɚ ɤ
.
ε+<<ε−
.
...,,,
29
ɉɭɫɬɶ
A
Ɍɨɝɞɚ
1>
, ɩɨɞɛɟɪɟɦ ε ɬɚɤ, ɱɬɨɛɵ
a
+
n
ɢɥɢ
11>
a
n
+1
.
12>=ε− AA
, ɬ. ɟ. ɩɨɫɥɟɞɭɸɳɢɣ ɱɥɟɧ ɪɹɞɚ (2.1) ɜɫɟɝɞɚ
aa >
nn
ɛɨɥɶɲɟ ɩɪɟɞɵɞɭɳɟɝɨ, ɡɧɚɱɢɬ, ɧɟɨɛɯɨɞɢɦɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ (ɬɟɨɪɟɦɚ 1.5),
ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɞɨɥɠɟɧ ɫɬɪɟɦɢɬɶɫɹ ɤ ɧɭɥɸ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ,
ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ.
ɉɪɢɦɟɪ 2.3.
ɂɫɫɥɟɞɨɜɚɬɶ ɪɹɞɵ ɧɚ ɫɯɨɞɢɦɨɫɬɶ:
∞
n
5
1)
2)
¦
n
¦
n
n
=
∞
=
1
;
()
+1!17
n
()
−⋅⋅⋅⋅
n
()
⋅+
!1
nn
12...753
.
Ɋɟɲɟɧɢɟ.
1. Ɍɚɤ ɤɚɤ
a
n
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɢɡɧɚɤɨɦ Ⱦɚɥɚɦɛɟɪɚ (ɬɟɨɪɟɦɚ 2.4):
n
n
a
n
a
lim lim
nn
→∞ →∞
, ɬɨ
()
!175+=n
1
+
n
5
=
1
+
n
+
a
n
1
+
n
() ()( )
+
1
=⋅=
nn
77 1! 2 5
⋅++
lim 0 1.
==<
→∞
n
72
n
=
55
⋅
nn
()( )
⋅
n
n
nnn
n
71!
5
n
⋅+
()
.
2!17755!27
++⋅
n
+
()
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɹɞ ɫɯɨɞɢɬɫɹ.
2. Ɉɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɢɦɟɟɬ ɜɢɞ
()
12...753
n
−⋅⋅⋅⋅
a
=
n
()
⋅+
,
!1
nn
30
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