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Файл:Числовые и функциональные ряды. Учебник
.pdf
Ʉɚɤ ɭɠɟ ɨɬɦɟɱɚɥɨɫɶ, ɪɚɡɧɨɫɬɢ ɜ ɫɤɨɛɤɚɯ ɩɨɥɨɠɢɬɟɥɶɧɵ, ɬ. ɟ.
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
{}
642
– ɦɨɧɨɬɨɧɧɨ ɭɛɵɜɚɸɳɚɹ. ɂɦɟɟɦ:
S
12 +n
ɢ
...
<<< SSS
>>> SSS
531
, ɩɪɢɱɟɦ
...
.
aSS
<<
1122
nn
+
ɢ
112aSn<+
ɂɬɚɤ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɪɹɞɚ (3.1) ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɫɜɨ-
ɟɦɭ ɩɪɟɞɟɥɭ ɤɨɥɟɛɥɹɫɶ ɢ ɫɭɦɦɚ ɪɹɞɚ ɦɟɧɶɲɟ ɟɝɨ ɩɟɪɜɨɝɨ ɱɥɟɧɚ
(ɪɢɫ. 3.1).
a
1
Ɋɢɫ. 3.1
ɉɨɫɤɨɥɶɤɭ ɨɫɬɚɬɨɤ ɪɹɞɚ (3.1)
aaR
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ
...
21
+++=mmm
ɬɟɨɪɟɦɵ, ɡɧɚɱɢɬ, ɞɥɹ ɧɟɝɨ ɜɵɩɨɥɧɹɸɬɫɹ ɞɨɤɚɡɚɧɧɵɟ ɜɵɲɟ ɡɚɤɥɸɱɟɧɢɹ, ɬ. ɟ.
. Ɍɪɟɬɶɟ ɭɬɜɟɪɠɞɟɧɢɟ ɬɚɤɠɟ ɞɨɤɚɡɚɧɨ.
aR
1
+<mm
ɉɪɢɦɟɪ 3.1.
ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɯɫɹ ɪɹɞɨɜ:
1)
2)
3)
¦
n
¦
n
¦
n
∞
1
=
∞
1
=
∞
1
=
1
n
+
()
−
1
n
n
()
⋅−
31
2
n
+
3
n
1
n
+
()
()
+
n
n
⋅−
;
!1
;
n
21
.
Ɋɟɲɟɧɢɟ.
1. ɑɥɟɧɵ ɪɹɞɚ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɭɛɵɜɚɸɬ
1
1
§
1
¨
111+>nn
2
©
1
·
>>>> ...
...
3
¸
n
¹
1
ɢ ɨɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ
lim =
0
, ɬ. ɟ. ɨɛɚ ɭɫɥɨɜɢɹ ɩɪɢɡɧɚɤɚ Ʌɟɣ-
∞→nn
ɛɧɢɰɚ (ɬɟɨɪɟɦɚ 3.1) ɜɵɩɨɥɧɹɸɬɫɹ, ɡɧɚɱɢɬ, ɞɚɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ.
∞
n
n
⋅−
31
.
2
n
+
3
n
2. ȼɵɱɢɫɥɢɦ ɩɪɟɞɟɥ ɨɛɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ:
¦
n
1
=
()
41
nnn
3 3 ln3 3 ln 3
lim lim lim
nnn
→∞ →∞ →∞
nnn
33ln323ln32
+⋅+⋅+
∞⋅ ⋅
ªº
== = =
22
«»
∞
nn
¬¼
2
3
n
3ln3
⋅
lim 1 0
==≠
n
→∞
3
n
3ln3
⋅
(ɡɞɟɫɶ ɬɪɢɠɞɵ ɢɫɩɨɥɶɡɨɜɚɧɨ ɩɪɚɜɢɥɨ Ʌɨɩɢɬɚɥɹ).
ȼɬɨɪɨɟ ɭɫɥɨɜɢɟ ɩɪɢɡɧɚɤɚ Ʌɟɣɛɧɢɰɚ (ɬɟɨɪɟɦɚ 3.1) ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɡɧɚɱɢɬ,
ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ.
3. ɉɪɨɜɟɪɢɦ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɣ ɩɪɢɡɧɚɤɚ Ʌɟɣɛɧɢɰɚ (ɬɟɨɪɟɦɚ 3.1). ɋɪɚɜ-
ɧɢɦ ɩɨɫɥɟɞɭɸɳɢɣ
ɢ ɩɪɟɞɵɞɭɳɢɣ
a
1+n
ɱɥɟɧɵ ɪɹɞɚ:
a
n
a
n
()
n
,
=
a
+
1
n
!12+=n
1
+
2
()()()()( )
++
22
⋅
=
!11
=
!2
+
nnn
22
⋅
,
2!1
+⋅+
nnnn
a
n
1
+
=
()( )
n
n
()
n
+
!1
⋅
22
⋅
nna
+⋅+
n
2!1
2
2
1
<
=
n
,
2
+
ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
ɉɪɟɞɟɥ ɨɛɳɟɝɨ ɱɥɟɧɚ ɪɹɞɚ
, ɬ. ɟ. ɩɟɪɜɨɟ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɹɟɬɫɹ.
aa
1+>nn
lim =
n
n
2
()
+
!1
n
∞→
0
(ɬɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɪɨɫɬɚ ɡɧɚɦɟɧɚ-
ɬɟɥɹ ɦɧɨɝɨ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ ɪɨɫɬɚ ɱɢɫɥɢɬɟɥɹ) ɨɛɚ ɭɫɥɨɜɢɹ ɩɪɢɡɧɚɤɚ ɜɵɩɨɥɧɹɸɬɫɹ, ɪɹɞ ɫɯɨɞɢɬɫɹ.
X
ɉɪɢɦɟɪ 3.2.
∞
n
n
⋅−
31
ɫ ɬɨɱɧɨɫɬɶɸ
n
+
3
n
.
01,0=δ
ȼɵɱɢɫɥɢɬɶ ɫɭɦɦɭ ɪɹɞɚ
()
¦
()
=
1
n
Ɋɟɲɟɧɢɟ.
ɑɥɟɧɵ ɪɹɞɚ ɭɛɵɜɚɸɬ ɫ ɜɨɡɪɚɫɬɚɧɢɟɦ ɧɨɦɟɪɚ:
n
3
>
n
()() ()
3
+
n
33
⋅
1
+
n
4
+
n
3
=
nnn
+
3
⋅
n
4
§
,
¨
4
+
n
©
·
.
<+1
¸
43n
¹
Ɉɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:
n
3
lim =
n
()
n
=
n
+
n
3
lim
3
§
·
0
¨
+
n
∞→∞→
©
.
¸
3
¹
42
Ɍɚɤ ɤɚɤ ɭɫɥɨɜɢɹ ɩɪɢɡɧɚɤɚ Ʌɟɣɛɧɢɰɚ (ɬɟɨɪɟɦɚ 3.1) ɜɵɩɨɥɧɹɸɬɫɹ, ɬɨ ɪɹɞ ɫɯɨ-
ɞɢɬɫɹ.
ȼɵɩɢɲɟɦ ɱɥɟɧɵ ɪɹɞɚ:
27
216
;
25
9
;
3
;
4
81
2401
;
54321
===== aaaaa
32768
243
.
ɇɚɱɢɧɚɹ ɫ ɩɹɬɨɝɨ ɱɥɟɧɵ ɪɹɞɚ ɦɟɧɶɲɟ
δ
()
.
01,05=δ<a
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɫɭɦɦɵ ɪɹɞɚ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ ɞɨɫɬɚ-
ɬɨɱɧɨ ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɩɟɪɜɵɟ ɱɟɬɵɪɟ ɱɥɟɧɚ ɪɹɞɚ:
∞
n
()
¦
=
1
n
n
3
31
⋅−
n
()
+
n
4
3
25
9
216
2401
−≈+−+≈
. X
4813,0
81
27
3.2. Ⱥɛɫɨɥɸɬɧɚɹ ɢ ɭɫɥɨɜɧɚɹ ɫɯɨɞɢɦɨɫɬɶ
ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ ɱɢɫɥɨɜɵɯ ɪɹɞɨɜ
ɉɭɫɬɶ ɱɥɟɧɵ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ
321
ɢɦɟɸɬ ɩɪɨɢɡɜɨɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɡɧɚɤɨɜ.
Ɋɚɫɫɦɨɬɪɢɦ ɨɞɢɧ ɨɛɳɢɣ ɞɨɫɬɚɬɨɱɧɵɣ ɩɪɢɡɧɚɤ ɫɯɨɞɢɦɨɫɬɢ ɡɧɚɤɨɩɟɪɟɦɟɧ-
ɧɨɝɨ ɪɹɞɚ.
Ɍɟɨɪɟɦɚ 3.2.
ɉɪɢɡɧɚɤ ɚɛɫɨɥɸɬɧɨɣ ɫɯɨɞɢɦɨɫɬɢ. ȿɫɥɢ ɞɥɹ ɡɧɚɤɨɩɟɪɟɦɟɧɧɨɝɨ ɪɹɞɚ (3.2)
ɫɯɨɞɢɬɫɹ ɪɹɞ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɟɝɨ ɱɥɟɧɨɜ
¦
n
∞
n
=
1
ɬɨ ɢ ɫɚɦ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɣ ɪɹɞ ɬɚɤɠɟ ɫɯɨɞɢɬɫɹ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ.
Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
ɜɫɟɯ ɩɨɥɨɠɢɬɟɥɶɧɵɯ ɱɥɟɧɨɜ, ɚ ɱɟɪɟɡ
ɪɢɰɚɬɟɥɶɧɵɯ ɱɥɟɧɨɜ ɫɪɟɞɢ ɩɟɪɜɵɯ
ɫɭɦɦɭ n ɩɟɪɜɵɯ ɱɥɟɧɨɜ ɪɹɞɚ (3.2), ɱɟɪɟɡ
S
n
−=
−
S
– ɫɭɦɦɭ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɜɫɟɯ ɨɬ-
n
n
ɱɥɟɧɨɜ ɪɹɞɚ. Ɍɨɝɞɚ
−+
SSS
ɢ
nnn
ɝɞɟ
21
aaa +++=σ ...
– ɫɭɦɦɚ n ɩɟɪɜɵɯ ɱɥɟɧɨɜ ɪɹɞɚ (3.3).
nn
∞
=++++
......
321
, (3.2)
aaaaa
nn
¦
=
1
n
, (3.3)
......
+++++=
aaaaa
n
S
−+
+=σ
SS
,
nnn
+
– ɫɭɦɦɭ
n
43
+
−
S
Ɍɚɤ ɤɚɤ ɩɨ ɭɫɥɨɜɢɸ
ɢɦɟɟɬ ɩɪɟɞɟɥ (ɨɛɨɡɧɚɱɢɦ ɟɝɨ ɱɟɪɟɡ σ), ɚ
σ
n
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɢ ɜɨɡɪɚɫɬɚɸɳɢɟ ɮɭɧɤɰɢɢ ɨɬ n, ɩɪɢɱɟɦ
−
ɫɬɪɟɦɢɬɫɹ ɤ ɩɪɟɞɟɥɭ, ɬ. ɟ. ɪɹɞ (3.2) ɫɯɨɞɢɬɫɹ.
, ɬɨ ɢ ɨɧɢ ɢɦɟɸɬ ɩɪɟɞɟɥɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ
nn
+
S
−+
ɩɪɢ
nnn
S
ɢ
n
n
σ<σ≤
nn
∞→n
Ɂɚɦɟɱɚɧɢɟ.
∞
ɉɪɢɡɧɚɤ ɚɛɫɨɥɸɬɧɨɣ ɫɯɨɞɢɦɨɫɬɢ ɧɟ ɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦ, ɬ. ɟ. ɪɹɞ
∞
ɦɨɠɟɬ ɫɯɨɞɢɬɶɫɹ ɢ ɬɨɝɞɚ, ɤɨɝɞɚ ɪɹɞ
¦
a
n
=
1n
ɪɚɫɯɨɞɢɬɫɹ.
¦
a
=
1n
Ɂɧɚɤɨɩɟɪɟɦɟɧɧɵɣ ɪɹɞ (3.2), ɞɥɹ ɤɨɬɨɪɨɝɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬ-
ɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɜɢɞɚ (3.3) ɫɯɨɞɢɬɫɹ, ɧɚɡɵɜɚɟɬɫɹ
ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɦɫɹ.
ȿɫɥɢ ɪɹɞ (3.2) ɫɯɨɞɢɬɫɹ, ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɪɹɞ (3.3) ɪɚɫɯɨɞɢɬɫɹ, ɬɨ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɣ ɪɹɞ ɧɚɡɵɜɚɟɬɫɹ
ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɢɦɫɹ.
ɉɪɢɦɟɪ 3.3.
ɂɫɫɥɟɞɨɜɚɬɶ ɯɚɪɚɤɬɟɪ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɨɜ ɩɪɢɦɟɪɚ 3.1.
Ɋɟɲɟɧɢɟ.
1. Ⱦɥɹ ɪɹɞɚ
ɦɨɧɢɱɟɫɤɢɣ ɪɹɞ
¦
n
+∞
¦
n
∞+
=
=1
+
1
n
()
−
1
ɪɹɞɨɦ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɹɜɥɹɟɬɫɹ ɝɚɪ-
n
1
1
, ɤɨɬɨɪɵɣ, ɤɚɤ ɢɡɜɟɫɬɧɨ, ɪɚɫɯɨɞɢɬɫɹ, ɬ. ɟ. ɢɫɯɨɞɧɵɣ ɪɹɞ ɫɯɨ-
n
ɞɢɬɫɹ ɭɫɥɨɜɧɨ.
2. Ⱦɥɹ ɪɹɞɚ
ɱɥɟɧɨɜ ɢɦɟɟɬ ɜɢɞ
¦
n
∞+
=
¦
n
+
1
n
()
()
+
n
1
∞+
n
2
()
+1!1
n
=
n
⋅−
21
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɪɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ
!1
.
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɩɪɢɡɧɚɤɨɦ Ⱦɚɥɚɦɛɟɪɚ (ɬɟɨɪɟɦɚ 2.4).
a
n
+
1
limlim
=
∞→
n
n
()( )
n
n
()
+
22
⋅
2!1
nna
+⋅+
!1
n
⋅
=
n
2
n
lim
2
n
+
∞→∞→
10
<=
2
.
Ɋɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɫɯɨɞɢɬɫɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɯɨɞɧɵɣ ɪɹɞ ɫɯɨ-
ɞɢɬɫɹ ɚɛɫɨɥɸɬɧɨ.
X
ɉɪɢɦɟɪ 3.4.
ɂɫɫɥɟɞɨɜɚɬɶ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ
∞
()
¦
1
n
=
n
§
n
−
1
¨
©
2
n
+
157
·
.
¸
−
114
n
¹
44
–
ɢ
n
Ɋɟɲɟɧɢɟ.
2
n
157
+
·
.
¸
114
n
−
¹
Ɋɹɞ ɹɜɥɹɟɬɫɹ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɦɫɹ ɫ ɨɛɳɢɦ ɱɥɟɧɨɦ
n
§
a
=
¨
n
©
ɉɪɨɜɟɪɢɦ ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɣ ɩɪɢɡɧɚɤɚ Ʌɟɣɛɧɢɰɚ (ɬɟɨɪɟɦɚ 3.1). Ɉɱɟ-
ɜɢɞɧɨ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɧɨɦɟɪɚ ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ
2
n
157
+
n
§
=
a
¨
n
©
·
¸
114
−
n
¹
=≥
a
+
1
n
227
+
n
§
¨
1314
+
n
©
2
()
1
+
n
·
¸
¹
,
n
ɬ. ɟ. ɱɥɟɧɵ ɪɹɞɚ ɭɛɵɜɚɸɬ ɫ ɜɨɡɪɚɫɬɚɧɢɟɦ
. Ɉɛɳɢɣ ɱɥɟɧ ɪɹɞɚ ɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ:
22
n
+
157
n
a
n
n
¨
n
©
§
=
limlim
·
lim
=
¸
−
n
114
n
¹
n
1
·
§
¸
¨
2
∞→∞→∞→
©
¹
.
0
=
Ɉɛɚ ɭɫɥɨɜɢɹ ɜɵɩɨɥɧɹɸɬɫɹ, ɪɹɞ ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ. ɉɪɨɜɟɪɢɦ ɚɛɫɨɥɸɬɧɭɸ
ɫɯɨɞɢɦɨɫɬɶ. Ɋɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɱɥɟɧɨɜ ɢɦɟɟɬ ɜɢɞ
2
∞
¦
n
n
§
¨
©
1
=
n
+
157
·
.
¸
−
114
n
¹
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɚɞɢɤɚɥɶɧɵɦ ɩɪɢɡɧɚɤɨɦ Ʉɨɲɢ (ɬɟɨɪɟɦɚ 2.5):
2
n
§
n
a
n
n
n
limlim
=
¨
n
©
n
157
+
·
=
¸
n
114
−
¹
lim
n
n
§
¨
©
n
157
+
·
lim
=
¸
n
114
−
n
¹
n
1
·
§
¨
2
∞→∞→∞→∞→
©
.
0
=
¸
¹
Ɋɹɞ ɢɡ ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ ɫɯɨɞɢɬɫɹ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɯɨɞɧɵɣ ɡɧɚɤɨɩɟɪɟ-
ɦɟɧɧɵɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɢ ɩɪɢɬɨɦ ɚɛɫɨɥɸɬɧɨ.
X
Ɋɚɡɝɪɚɧɢɱɟɧɢɟ ɚɛɫɨɥɸɬɧɨɣ ɢ ɧɟɚɛɫɨɥɸɬɧɨɣ (ɭɫɥɨɜɧɨɣ) ɫɯɨɞɢɦɨɫɬɟɣ ɪɹɞɨɜ
ɹɜɥɹɟɬɫɹ ɜɟɫɶɦɚ ɫɭɳɟɫɬɜɟɧɧɵɦ, ɬɚɤ ɤɚɤ ɧɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ ɤɨɧɟɱɧɵɯ ɫɭɦɦ ɩɟɪɟɧɨɫɹɬɫɹ ɬɨɥɶɤɨ ɧɚ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ ɷɬɢɦɢ ɫɜɨɣɫɬɜɚɦɢ ɧɟ ɨɛɥɚɞɚɸɬ.
Ɍɚɤ, ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ ɨɛɥɚɞɚɸɬ, ɤɚɤ ɢ ɨɛɵɱɧɵɟ ɫɭɦɦɵ ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɫɥɚɝɚɟɦɵɯ, ɩɟɪɟɦɟɫɬɢɬɟɥɶɧɵɦ ɫɜɨɣɫɬɜɨɦ
, (
ɬ. ɟ. ɩɪɢ ɥɸɛɨɣ ɩɟɪɟɦɟɧɟ
ɦɟɫɬ ɱɥɟɧɨɜ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɪɹɞɚ ɨɧ ɨɫɬɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɦɫɹ ɢ
ɫ ɬɨɣ ɠɟ ɫɭɦɦɨɣ). ɗɬɢɦ ɫɜɨɣɫɬɜɨɦ ɧɟ ɨɛɥɚɞɚɸɬ ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ. Ɉɤɚɡɵɜɚɟɬɫɹ, ɩɟɪɟɫɬɚɜɥɹɹ ɱɥɟɧɵ ɬɚɤɨɝɨ ɪɹɞɚ, ɦɨɠɧɨ ɞɨɛɢɬɶɫɹ ɬɨɝɨ, ɱɬɨ ɫɭɦɦɚ ɪɹɞɚ
ɢɡɦɟɧɢɬɫɹ.
45
ɇɚɩɪɢɦɟɪ, ɩɟɪɟɫɬɚɜɢɦ ɱɥɟɧɵ ɭɫɥɨɜɧɨ ɫɯɨɞɹɳɟɝɨɫɹ ɪɹɞɚ
1
1
1
1
1
1
1
1 +−+−+−+−
2
ɬɚɤ, ɱɬɨɛɵ ɩɨɫɥɟ ɤɚɠɞɨɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɱɥɟɧɚ ɫɬɨɹɥɢ ɞɜɚ ɨɬɪɢɰɚɬɟɥɶɧɵɯ:
1 +−−+−−
Ɂɚɬɟɦ ɫɥɨɠɢɦ ɤɚɠɞɵɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɱɥɟɧ ɫ ɢɞɭɳɢɦ ɩɨɫɥɟ ɧɟɝɨ ɨɬɪɢɰɚ-
ɬɟɥɶɧɵɦ:
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɪɹɞ, ɱɥɟɧɵ ɤɨɬɨɪɨɝɨ ɨɛɪɚɡɨɜɚɧɵ ɩɪɨɢɡɜɟɞɟɧɢɹɦɢ
ɱɥɟɧɨɜ ɢɫɯɨɞɧɨɝɨ ɪɹɞɚ ɧɚ
ɞɢɬɫɹ ɢ ɫɭɦɦɚ ɟɝɨ ɪɚɜɧɚ ɩɨɥɨɜɢɧɟ ɫɭɦɦɵ ɢɫɯɨɞɧɨɝɨ ɪɹɞɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡɦɟɧɢɜ ɬɨɥɶɤɨ ɩɨɪɹɞɨɤ ɫɥɟɞɨɜɚɧɢɹ ɱɥɟɧɨɜ ɪɹɞɚ, ɦɵ ɭɦɟɧɶɲɢɥɢ ɟɝɨ ɫɭɦɦɭ ɜɞɜɨɟ.
Ɉɬɦɟɬɢɦ ɬɚɤɠɟ, ɱɬɨ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɟɫɹ ɪɹɞɵ ɨɛɥɚɞɚɸɬ ɟɳɟ ɨɞɧɢɦ ɜɚɠɧɵɦ ɫɜɨɣɫɬɜɨɦ: ɢɯ ɦɨɠɧɨ ɩɟɪɟɦɧɨɠɚɬɶ. ɉɪɢ ɷɬɨɦ ɩɨɥɭɱɚɟɬɫɹ ɪɹɞ
ɧɵɯ ɩɚɪɧɵɯ ɩɪɨɢɡɜɟɞɟɧɢɣ ɱɥɟɧɨɜ ɢɫɯɨɞɧɵɯ ɪɹɞɨɜ, ɤɨɬɨɪɵɣ ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɹɳɢɦɫɹ, ɢ ɟɝɨ ɫɭɦɦɚ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɫɭɦɦ ɪɹɞɨɜ ɫɨɦɧɨɠɢɬɟɥɟɣ.
1
2
4
3
1
1
4
2
1
1
4
2
. ɉɨ ɫɜɨɣɫɬɜɭ (1.3) ɢɦɟɟɦ, ɱɬɨ ɷɬɨɬ ɪɹɞ ɬɨɠɟ ɫɯɨ-
6
5
1
613
1
1
...
+−+−
8
6
...
8
7
1
...
.
8
.
ɢɡ ɜɫɟɜɨɡɦɨɠ-
46
ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
3.1. ɂɫɫɥɟɞɨɜɚɬɶ ɞɚɧɧɵɟ ɪɹɞɵ ɧɚ ɚɛɫɨɥɸɬɧɭɸ ɢɥɢ ɭɫɥɨɜɧɭɸ ɫɯɨɞɢɦɨɫɬɶ:
1)
3)
5)
7)
9)
11)
∞
¦
=
1
ɩ
∞
¦
=
1
ɩ
∞
¦
=
1
ɩ
∞
¦
=
1
ɩ
∞
()
¦
=
1
ɩ
∞
¦
=
ɩ
()
1
1
ɩ
+
()
−
1
; 2)
()
+
1ln
n
1
n
−
()
−
1
; 4)
()
ɩ
()( )
−
−
()
ɩ
⋅+
41
n
ɩ
−⋅−
31
2
ɩ
ɩ
1
−
ɩ
1
ɩ
1
−
; 6)
−
1
ɩ
§
+
1
¨
+
12
ɩ
©
1
n
arctg
; 12)
2
+
1
n
ɩ
·
; 8)
¸
¹
−⋅⋅⋅⋅
)23(...741
ɩ
; 10)
+⋅⋅⋅⋅
)52(...1197
ɩ
3.2. ȼɵɱɢɫɥɢɬɶ ɫɭɦɦɭ ɞɚɧɧɵɯ ɪɹɞɨɜ ɫ ɭɤɚɡɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ
1)
3)
5)
7)
¦
ɩ
¦
ɩ
¦
ɩ
¦
ɩ
∞
=
1
∞
=
1
∞
=
1
∞
=
1
1
n
+
()
1
−
2
3
n
1
n
+
()
1
−
n
!
ɩ
()
1
−
ɩ
ɩ
()
−
1
2
01,0,
=δ
01,0,
=δ
ɩ
12
+
,
3
()
1
+
ɩɩ
,
ɩ
1,0=δ
; 2)
; 4)
01,0=δ
; 6)
; 8)
∞
ɩ
()
1
−
¦
ln
nɩ
=
1
ɩ
∞
n
()
1
−
¦
1
ɩ
+
=
1
ɩ
∞
ɩ
()
−
1
¦
=
1
ɩ
∞+
n
()
1
¦
=
1
n
∞+
ɩ
()
−
1
¦
=
1
ɩ
∞
ɩ
()
1
−
¦
=
1
ɩ
∞
n
()
1
−
¦
()
2
ɩ
=
1
ɩ
∞
()
1
−
¦
()
ɩn
=
1
ɩ
∞+
ɩ
()
1
−
¦
=
1
ɩ
∞
ɩ
()
−
1
¦
=
1
ɩ
;
ɩ
2
·
§
⋅
¸
¨
3
¹
©
ɩ
+
1
()
ɩɩ
−
n
§
⋅−
¨
+
n
©
+
1
1
+
3
n
12!
+
ɩ
+
+
1
4
+
ɩ
ɩ
+
1
,
ɩ
3
;
+
12
;
+⋅
1
n
23
·
;
¸
15
¹
+⋅⋅⋅⋅
)12(...753
ɩ
;
+⋅⋅⋅⋅
)23(...1185
ɩ
3
.
2
+⋅
)1(arctg
nn
δ :
;
001,0,
=δ
;
001,0,
=δ
3
,
01,0=δ
;
2
1,0=δ
.
47
ɌȿɋɌɕ ȾɅə ɋȺɆɈɄɈɇɌɊɈɅə
ɂ ɋɂɋɌȿɆȺɌɂɁȺɐɂɂ ɁɇȺɇɂɃ
ȼɚɪɢɚɧɬ 1
Ɂɚɞɚɧɢɟ 1
Ɂɚɞɚɧ ɱɢɫɥɨɜɨɣ ɪɹɞ ɫ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ. ȼɟɪɧɵɦɢ ɭɬɜɟɪɠɞɟɧɢɹɦɢ
ɹɜɥɹɸɬɫɹ:
Ⱥ: ɱɢɫɥɨɜɨɣ ɪɹɞ ɫɯɨɞɢɬɫɹ ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ, ɤɨɝɞɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ
ɟɝɨ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɨɝɪɚɧɢɱɟɧɚ ɫɜɟɪɯɭ;
B: ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɫɯɨɞɹɳɟɝɨɫɹ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɦɨɧɨɬɨɧɧɚ;
C: ɫɭɳɟɫɬɜɭɟɬ ɪɚɫɯɨɞɹɳɢɣɫɹ ɱɢɫɥɨɜɨɣ ɪɹɞ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ;
D: ɜɫɟ ɱɢɫɥɨɜɵɟ ɪɹɞɵ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ ɱɥɟɧɚɦɢ ɫɯɨɞɹɬɫɹ.
1) A ɢ D;
2) B ɢ C;
3) A ɢ C;
4) A ɢ B;
5) C ɢ D.
Ɂɚɞɚɧɢɟ 2
ȼɟɪɧɵɦɢ ɭɬɜɟɪɠɞɟɧɢɹɦɢ, ɫɪɟɞɢ ɩɪɢɜɟɞɟɧɧɵɯ, ɹɜɥɹɸɬɫɹ:
A: ɨɛɳɢɣ ɦɧɨɠɢɬɟɥɶ ɧɟɥɶɡɹ ɜɵɧɨɫɢɬɶ ɡɚ ɡɧɚɤ ɫɭɦɦɵ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ;
B: ɫɯɨɞɹɳɢɟɫɹ ɱɢɫɥɨɜɵɟ ɪɹɞɵ ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɫɤɥɚɞɵɜɚɬɶ;
C: ɫɭɳɟɫɬɜɭɟɬ ɫɯɨɞɹɳɢɣɫɹ ɱɢɫɥɨɜɨɣ ɪɹɞ,
D: ɜ ɫɯɨɞɹɳɟɦɫɹ ɱɢɫɥɨɜɨɦ ɪɹɞɭ ɦɨɠɧɨ ɩɪɨɢɡɜɨɥɶɧɨ ɝɪɭɩɩɢɪɨɜɚɬɶ ɟɝɨ
ɱɥɟɧɵ, ɛɟɡ ɢɡɦɟɧɟɧɢɹ ɢɯ ɩɨɪɹɞɤɚ.
1) B ɢ C;
2) B ɢ D;
3) C ɢ D;
4) A ɢ B;
5) B, C, D.
n
-ɨɫɬɚɬɨɤ ɤɨɬɨɪɨɝɨ ɪɚɫɯɨɞɢɬɫɹ;
Ɂɚɞɚɧɢɟ 3
Ɂɚɞɚɧ ɱɢɫɥɨɜɨɣ ɪɹɞ
A:
+∞→nn
B: ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɪɚɫɯɨɞɢɬɫɹ;
C: ɫɭɦɦɚ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɤɨɧɟɱɧɚ;
D: ɱɢɫɥɨɜɨɣ ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ;
E: ɩɪɟɞɟɥ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɱɢɫɥɨɜɨɝɨ ɪɹɞɚ ɧɟ ɫɭɳɟ-
ɫɬɜɭɟɬ.
∞=
alim
∞+
¦
=
1
n
;
n
+
()
−
1
1
. ȼɟɪɧɵɦ ɭɬɜɟɪɠɞɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɬɨɥɶɤɨ:
48
1) A, C, E;
2) B ɢ E;
3) B, D, E;
4) D ɢ E;
5) B, C, D.
Ɂɚɞɚɧɢɟ 4
Ⱦɥɹ ɪɹɞɨɜ
∞+
1
A:
B:
C:
D:
¦
n
¦
n
¦
¦
=
∞+
sin
=
1
∞+
n
n
=
12n
3
n
;
()
+21ln
n
1
;
n
;
1
ɜɵɛɪɚɬɶ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ.
2
+
1
1) ɬɨɥɶɤɨ A, B, D – ɪɚɫɯɨɞɹɬɫɹ;
2) ɬɨɥɶɤɨ A ɢ B – ɪɚɫɯɨɞɹɬɫɹ;
3) ɬɨɥɶɤɨ B ɢ C – ɪɚɫɯɨɞɹɬɫɹ;
4) A, B, D – ɪɚɫɯɨɞɹɬɫɹ;
5) ɜɫɟ ɪɚɫɯɨɞɹɬɫɹ.
Ɂɚɞɚɧɢɟ 5
∞+
Ⱦɥɹ ɪɹɞɚ
A:
+∞→nn
1
ɜɵɛɪɚɬɶ ɜɟɪɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ:
¦
4
n
=
1
n
;
0lim =
a
B: ɪɹɞ ɪɚɫɯɨɞɢɬɫɹ;
C: ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɪɚɜɧɨɫɢɥɶɧɚ ɫɯɨɞɢɦɨɫɬɢ ɢɧɬɟɝɪɚɥɚ
D:
lim =
+∞→
n
n
a
.
1
n
a
1
+
1) ɬɨɥɶɤɨ C ɢ D;
2) ɬɨɥɶɤɨ A ɢ C;
3) ɬɨɥɶɤɨ A ɢ B;
4) ɬɨɥɶɤɨ A ɢ D;
5) ɬɨɥɶɤɨ A, C, D.
∞+
dx
;
³
4
x
1
49
Ɂɚɞɚɧɢɟ 6
∞+
ɉɭɫɬɶ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɣ ɱɢɫɥɨɜɨɣ ɪɹɞ
ɫɯɨɞɢɬɫɹ. Ʉɚɤɢɟ ɢɡ ɭɫɥɨɜɢɣ
a
n
¦
=
1n
ɹɜɥɹɸɬɫɹ ɩɪɢɡɧɚɤɚɦɢ ɫɯɨɞɢɦɨɫɬɢ?
n
qa
1)
2)
lim >=
+∞→
n
3)
+∞→
n
4)
+∞→nn
ɢ ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɝɨ ɧɨɦɟɪɚ
()
1;0∈∃ q
a
1
+
n
a
n
n
a
n
0lim =
a
;
1
D
1lim =λ=
;
;
5) ɧɚɱɢɧɚɹ ɫ ɧɟɤɨɬɨɪɨɝɨ ɧɨɦɟɪɚ
n
1>
a
.
n
≤
;
n
Ɂɚɞɚɧɢɟ 7
Ɉɬɦɟɬɶɬɟ ɜɟɪɧɵɟ ɭɬɜɟɪɠɞɟɧɢɹ:
A: ɥɸɛɨɣ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɣɫɹ ɪɹɞ ɫɯɨɞɢɬɫɹ;
B: ɪɹɞ
¦
n
∞+
=
1
()
1
−
n
1
ɫɯɨɞɢɬɫɹ;
1
−
n
C: ɜ ɭɫɥɨɜɢɹɯ ɬɟɨɪɟɦɵ Ʌɟɣɛɧɢɰɚ ɫɭɦɦɚ ɪɹɞɚ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ
≤≤nnSSS
.
122 +
1) A ɢ C;
2) A ɢ B;
3) ɬɨɥɶɤɨ B;
4) B ɢ C;
5) ɬɨɥɶɤɨ C.
Ɂɚɞɚɧɢɟ 8
Ⱦɥɹ ɪɹɞɨɜ
¦
∞+
a
n
=
11n
()
ɢ
¦
∞+
ɜɵɛɪɚɬɶ ɜɟɪɧɵɟ ɭɬɜɟɪɠɞɟɧɢɹ:
()
a
n
=
12n
A: ɪɹɞ (1) ɫɯɨɞɢɬɫɹ ɭɫɥɨɜɧɨ, ɟɫɥɢ ɨɧ ɫɯɨɞɢɬɫɹ, ɚ ɪɹɞ (2) ɪɚɫɯɨɞɢɬɫɹ;
B: ɢɡ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ (1) ɫɥɟɞɭɟɬ ɟɝɨ ɚɛɫɨɥɸɬɧɚɹ ɫɯɨɞɢɦɨɫɬɶ;
C: ɟɫɥɢ ɪɹɞ (1) ɚɛɫɨɥɸɬɧɨ ɫɯɨɞɢɬɫɹ, ɬɨ ɫɯɨɞɢɬɫɹ (2);
D: ɢɡ ɫɯɨɞɢɦɨɫɬɢ (1), ɫɥɟɞɭɟɬ ɫɯɨɞɢɦɨɫɬɶ (2).
1) B ɢ D;
2) A ɢ C;
3) A ɢ B;
4) A ɢ D;
5) B ɢ C.
50
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