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Файл:Числовые и функциональные ряды. Учебник
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1
ɱɥɟɧɚ ɪɚɜɟɧ ɧɭɥɸ:
lim =
n
0
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ (ɬɟɨ-
n
∞→
ɪɟɦɚ 3.1) ɪɹɞ ɫɯɨɞɢɬɫɹ, ɧɨ ɷɬɚ ɫɯɨɞɢɦɨɫɬɶ ɹɜɥɹɟɬɫɹ ɭɫɥɨɜɧɨɣ, ɩɨɫɤɨɥɶɤɭ ɪɹɞ ɢɡ
∞+
ɚɛɫɨɥɸɬɧɵɯ ɜɟɥɢɱɢɧ
ɡɨɦ, ɪɹɞ ɞɥɹ ɮɭɧɤɰɢɢ
1
ɹɜɥɹɟɬɫɹ ɪɚɫɯɨɞɹɳɢɦɫɹ ɝɚɪɦɨɧɢɱɟɫɤɢɦ. Ɍɚɤɢɦ ɨɛɪɚ-
¦
n
1
n
=
ɫɯɨɞɢɬɫɹ ɩɪɢ
()
x+1ln
. X
(
]
1;1−∈x
Ɋɚɡɥɨɠɟɧɢɟ Ɉɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ
53
!5!3
!4!2
1
−
mm
21
⋅
53
53
−
k
−
k
k
x
()
!2
k
1
2
!!3
+
5!!4
⋅
32
!3!2
()
432
432
+
x
32
x
1
!!5
x
1
xe
xx
sin
1cos
x
() ()
m
()
arctg
xx
1ln
11
1
1
+
x
()
++=+ x
mxx
xx
∞
x
=
xsh
¦
()
=
1
k
∞
1
xch
¦
=
k
1
−
1
x
xxarc
sin
+
+=
3!!2
⋅
n
xxx
...
1...
−+−+−=
()
...
++++++=
!
n
12
−
n
xxx
1
+
n
()
n
!12
−
242
n
xxx
n
1...
+−+−+−=
()
!2
n
xxxx
1
−
n
n
()( )
()
1...
−+−+−=
()
...
7!!6
⋅
−⋅−
mmm
321
⋅⋅
12
−
n
xxx
1
−
n
n
12
−
n
n
+−++−+−=
xxxx
5312
xx
+++=
...
!5!3!12
422
xx
+++=+=
...
!4!2
∞
nn
=+++++=
......1
xxxx
¦
=
0
n
()
−
++
()( )
Rx∈
...
+
...
n
...1...
+−++−+−=+
[]
21
32
...
+
x
(
()
...
+
...1...1
12753
+
n
xnxxx
!!12
...
+
nn
12!!2
+⋅
Rx∈
Rx∈
(
]
1,1∈x
0; ɟɫɥɢ,1,1
≥−∈
]
mx
[]
1,1−∈x
()
1,1−∈x
()
∞<x
()
∞<x
1,<x
()
1,1−∈∀x
;01 ɟɫɥɢ,1,1-
<<−∈
mx
1ɟɫɥɢ,1,1
−≤−∈
3. Ɇɟɬɨɞ ɤɨɦɛɢɧɢɪɨɜɚɧɢɹ ɢɡɜɟɫɬɧɵɯ ɪɹɞɨɜ. Ɋɚɡɥɨɠɟɧɢɟ ɮɭɧɤɰɢɣ ɜ ɫɬɟ-
ɩɟɧɧɨɣ ɪɹɞ ɦɨɠɧɨ ɩɨɥɭɱɚɬɶ ɤɨɦɛɢɧɢɪɨɜɚɧɢɟɦ ɢɡɜɟɫɬɧɵɯ ɪɚɡɥɨɠɟɧɢɣ.
101
ɉɪɢɦɟɪ 6.7.
x
x
x
x
+
1
()
xf
=
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ ɩɨ ɫɬɟɩɟɧɹɦ
ɫɯɨɞɢɦɨɫɬɢ ɩɨɥɭɱɟɧɧɨɝɨ ɪɹɞɚ.
Ɋɟɲɟɧɢɟ.
ɂɦɟɟɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ
ɮɭɧɤɰɢɸ
ln
−
1
x
+
1
ln
−
1
()()
−−+=
1ln1ln
xx
.
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɟɡɭɥɶɬɚɬɨɦ ɪɚɡɥɨɠɟɧɢɹ ɮɭɧɤɰɢɢ ɢɡ ɩɪɢɦɟɪɚ 6.6:
()
1ln
ɝɞɟ
Ɂɚɩɢɲɟɦ ɪɹɞ ɞɥɹ
ɗɬɨɬ ɪɹɞ ɛɭɞɟɬ ɫɯɨɞɢɬɫɹ ɩɪɢ
.
(
]
1`;1−∈x
()
1ln
xx
, ɡɚɦɟɧɢɜ x ɧɚ
()
x−1ln
xx
432
...
++−+−=+
432
−
432
432
. Ɍɨɝɞɚ
()
1;1−∈x
1
−
n
()
xxxx
1
⋅−
n
, ɩɨɥɭɱɢɦ
n
xxxx
+−−−−−−=−
n
n
......
234 234
§·§·
+
1
xxxx xxx
=−+−+−−−−−=
ɝɞɟ
ln ... ...
1 234 234
(
]
1`;1−∈x
. X
xx
¨¸¨¸
−
x
©¹©¹
35 21
§·
=⋅ + + + + +
2 ... ... ,
xx x
x
¨¸
35 21
©¹
n
−
−
n
x
. ɍɤɚɡɚɬɶ ɨɛɥɚɫɬɶ
...,
+
.
102
ɁȺȾȺɇɂə ȾɅə ɋȺɆɈɋɌɈəɌȿɅɖɇɈȽɈ Ɋȿɒȿɇɂə
x
x
x
(
)
6.1. Ɋɚɡɥɨɠɢɬɶ ɞɚɧɧɵɟ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɭɤɚɡɚɧɧɵɯ ɬɨ-
ɱɟɤ
. ɍɤɚɡɚɬɶ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɩɨɥɭɱɟɧɧɨɝɨ ɪɹɞɚ:
x
0
1)
3)
5)
7)
x
()
)(
= x
xf
()
2
+
)(
= x
xf
2
0,2)(
== xxf
; 2)
0
; 4)
1,2ln)(
−=+= xxxf
0
1
2
x
1
+
ɯ
1,
−=
; 6)
0
1,
−=
; 8)
0
6.2. ɇɚɣɬɢ ɩɟɪɜɵɟ ɩɹɬɶ ɱɥɟɧɨɜ ɪɚɡɥɨɠɟɧɢɹ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ
ɥɨɪɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɭɤɚɡɚɧɧɵɯ ɬɨɱɟɤ
3
1)
3)
5)
7)
9)
x
2
−
3
3
0,)(
== xexxf
; 2) 0,)(
0
510
xx
xexf
0
== xxxxf
0
0
=+−= xxxxf
0
1,)(
==
; 6) 2,)(
2,)(
; 8)
1,ln)(
== xxxxf
; 10)
:
x
0
1,13)(
; 4)
6.3.
Ɋɚɡɥɨɠɢɬɶ ɞɚɧɧɵɟ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɇɚɤɥɨɪɟɧɚ ɩɨ ɫɬɟɩɟɧɹɦ
ɢɡɜɟɫɬɧɵɟ ɪɚɡɥɨɠɟɧɢɹ, ɢ ɭɤɚɡɚɬɶ ɨɛɥɚɫɬɢ ɫɯɨɞɢɦɨɫɬɢ ɩɨɥɭɱɟɧɧɵɯ ɪɹɞɨɜ:
22
1)
sin
; 2)
−
ɯ
()
1
)(
= x
xf
()
3
+
3
2
x
2
+
()()
;
xxcos
0,3)(
==
xxf
;
0
0
0
2
x
−=+= xɯxf
0
== xxexf
;
0
612
2
xx
xexf
0
4,)(
== xɯɯxf
;
0
2
;
2,3ln)(
−=+= xxxf
2,
−=
;
1,2)(
.
ɜ ɪɹɞ Ɍɟɣ-
()
xf
−=−+= xxxxf
0
−==
;
, ɢɫɩɨɥɶɡɭɹ
1,65)(
0
;
1,2ln2)(
−=+⋅+= xxxxf
.
3)
5)
7)
; 4)
x21 +
1
; 6)
x
e
25
1ln xx +
; 8)
e
1 x
x
11x−
−
;
2
4
x
;
·
§
¨
©
.
+51ln
¸
¹
6.4. ɉɪɟɞɫɬɚɜɢɬɶ ɞɚɧɧɵɟ ɢɧɬɟɝɪɚɥɵ ɜ ɜɢɞɟ ɪɹɞɨɜ ɩɨ ɫɬɟɩɟɧɹɦ x:
ɯ
dx
;
³
ɯ
ɟ
0
1)
2
2
−ɯɯ
; 2)
³
0
dxɟɯ
103
3)
5)
7)
9)
³
0
ɯ
³
0
ɯ
³
0
ɯ
³
0
dx
3
cos
; 4)
3
+ɯx
1
2
⋅
arctg
3
; 8)
dxɯ
()
+
1ln
x
dx
x
5
+ɯdxx
1
³
0
ɯ
; 6)
dxxx
; 10)
³
0
ɯ
³
0
arctg
⋅
sin
³
0
;
2
x
;
dx
x
2
;
dxxx
25
()
+ɯdxxx
1ln
.
104
Ƚɥɚɜɚ 7
ɉɊɂɅɈɀȿɇɂȿ ɑɂɋɅɈȼɕɏ ɊəȾɈȼ
Ʉ ɉɊɂȻɅɂɀȿɇɇɕɆ ȼɕɑɂɋɅȿɇɂəɆ
ɑɢɫɥɨɜɵɟ ɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɪɹɞɵ ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɜ ɩɪɢɛɥɢɠɟɧɧɵɯ
ɜɵɱɢɫɥɟɧɢɹɯ. Ɋɚɫɫɦɨɬɪɢɦ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɟ ɢɡ ɷɬɢɯ ɩɪɢɦɟɧɟɧɢɣ.
7.1. ȼɵɱɢɫɥɟɧɢɟ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɣ
ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ
ɩɪɢ
()
xf
ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɮɭɧɤɰɢɸ ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɜ ɫɬɟɩɟɧɧɨɣ
ɪɹɞ.
() () ()
10
n
n
......
+−++−+=
axaaxaaxf
ɧɚ ɢɧɬɟɪɜɚɥɟ
()
ɢ ɱɬɨ ɬɨɱɤɚ
RaRa +− ,
ɩɪɢɧɚɞɥɟɠɢɬ ɞɚɧɧɨɦɭ ɢɧɬɟɪɜɚɥɭ.
xx =
0
Ɍɨɝɞɚ
() ()() ()
2
020100
axaaxaaxaaxf
0
n
ȼɡɹɜ ɞɨɫɬɚɬɨɱɧɨɟ ɱɢɫɥɨ ɩɟɪɜɵɯ ɱɥɟɧɨɜ ɪɹɞɚ, ɬ. ɟ. ɧɟɤɨɬɨɪɭɸ ɱɚɫɬɢɱɧɭɸ
ɫɭɦɦɭ
, ɩɨɥɭɱɢɦ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ
()
xS
n
()() ()() ()
2
0201000
...
nn
ɬɨɱɧɨɫɬɶ ɤɨɬɨɪɨɝɨ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɜɨɡɪɚɫɬɚɧɢɟɦ
ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ, ɬ. ɟ.
() ()
xSxf
−
n
n
. Ⱥɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ
, ɪɚɜɧɚ ɦɨɞɭɥɸ ɨɫɬɚɬɤɚ ɪɹɞɚ:
00
() () ()
xrxSxf
=−
,
000
nn
1
+
() ( ) ( )
ɝɞɟ
+
nn
n
010
n
ɀɟɥɚɹ ɜɵɱɢɫɥɢɬɶ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ
n
ɜɡɹɬɶ ɫɭɦɦɭ ɬɚɤɨɝɨ ɱɢɫɥɚ
ɩɟɪɜɵɯ ɱɥɟɧɨɜ, ɱɬɨɛɵ
02
+
axaaxaxr
2
+
n
...
+−+−=
.
ɫ ɬɨɱɧɨɫɬɶɸ 0>ε , ɦɵ ɞɨɥɠɧɵ
()
xf
0
() () ()
nn
ε<=−
xrxSxf
.
000
ȿɫɥɢ ɮɭɧɤɰɢɹ ɪɚɡɥɨɠɟɧɚ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ Ɍɟɣɥɨɪɚ (ɢɥɢ Ɇɚɤɥɨɪɟɧɚ), ɬɨ ɜ
ɷɬɨɦ ɫɥɭɱɚɟ ɚɛɫɨɥɸɬɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɪɚɜɧɚ ɦɨɞɭɥɸ ɨɫɬɚɬɨɱɧɨɝɨ ɱɥɟɧɚ ɪɹɞɚ Ɍɟɣɥɨɪɚ (ɢɥɢ Ɇɚɤɥɨɪɟɧɚ):
xx =
0
n
......
+−++−+−+=
axaaxaaxaaxSxf −++−+−+=≈
0
ɫ ɡɚɞɚɧɧɨɣ
.
n
,
105
n
x
x
x
+
1
()
() () ()
==−
xRxSxf
nn
000
cf
()
n
()
+
!1
−⋅
1
+
n
xx
,
0
ɝɞɟ
∈
.
()
xxc ;
0
ɉɪɢɦɟɪ 7.1.
ȼɵɱɢɫɥɢɬɶ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
ɱɢɫɥɨ e.
001,0
Ɋɟɲɟɧɢɟ.
Ⱦɥɹ ɥɸɛɨɝɨ
ɢɦɟɟɬ ɦɟɫɬɨ ɪɚɡɥɨɠɟɧɢɟ (6.9)
x
1
xe
32
!3!2
n
xxx
...
...
++++++=
.
!
n
ɉɪɢ
1=
ɩɨɥɭɱɢɦ
1
11 ++++++=ne
!31!2
1
.
...
...
!
ȼɡɹɜ ɩɟɪɜɵɟ
1+n ɱɥɟɧɨɜ, ɩɨɥɭɱɢɦ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ
1
11 ++++++≈ne
!31!2
1
.
...
...
!
Ɉɰɟɧɢɦ ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɢɛɥɢɠɟɧɢɹ ɫ ɩɨɦɨɳɶɸ ɨɫɬɚɬɨɱɧɨɝɨ ɱɥɟɧɚ ɪɹɞɚ Ɇɚ-
ɤɥɨɪɟɧɚ. Ɍɚɤ ɤɚɤ
()
+1
()
exf =
xn
, ɬɨ
c
e
()
=
xR ,
n
()
+
n
1!1+
n
⋅
x
ɝɞɟ
ɉɪɢ
.
()
xc ;0∈
1=
ɢɦɟɟɦ
c
e
n
()
1
=
R
n
()
+
n
+
1
!1
c
e
1
=⋅
()
+
n
, 10 << c .
!1
Ɍɚɤ ɤɚɤ
ɉɪɢ
1
c
<< ee
3
5=n
()
!15
+
, ɩɨɥɭɱɚɟɦ
3
3
240
!6
()
R
n
1
001,0
>==
1+<
()
n
, ɚ ɩɪɢ
3
.
!1
3
6=n
()
+
1
<=
1680
!16
.
001,0
106
ɉɨɷɬɨɦɭ ɞɥɹ ɞɨɫɬɢɠɟɧɢɹ ɬɪɟɛɭɟɦɨɣ ɬɨɱɧɨɫɬɢ ɞɨɫɬɚɬɨɱɧɨ ɜɡɹɬɶ
x
x
ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
001,0
ɢɦɟɟɦ
. ɂɬɚɤ,
6=n
1
11 ++++++≈e
.
!61!51!41!31!2
Ʉɚɠɞɨɟ ɫɥɚɝɚɟɦɨɟ ɜɵɩɢɲɟɦ ɫ ɨɞɧɢɦ ɡɚɩɚɫɧɵɦ ɡɧɚɤɨɦ, ɱɬɨɛɵ ɤ ɧɚɲɟɣ
ɨɲɢɛɤɟ ɧɟ ɞɨɛɚɜɥɹɥɢɫɶ ɨɲɢɛɤɢ ɨɬ ɨɤɪɭɝɥɟɧɢɹ ɫɥɚɝɚɟɦɵɯ:
.
7181,20014,00083,00417,01667,05000,00000,10000,1 =++++++≈e
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
ɧɚɯɨɞɢɦ
001,0
. X
718,2=e
ɉɪɢɦɟɪ 7.2.
ȼɵɱɢɫɥɢɬɶ
Ɋɟɲɟɧɢɟ.
Ⱦɥɹ
sin ɢɦɟɟɦ ɪɚɡɥɨɠɟɧɢɟ (6.11), ɫɩɪɚɜɟɞɥɢɜɨɟ ɩɪɢ ɜɫɟɯ ɡɧɚɱɟɧɢɹɯ
0
ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
18sin
.
0001,0
:
1
−
n
()
1
+−+−=
()
n
sin
53
xx
...
!5!3
ɉɟɪɟɜɟɞɟɦ
0
ɜ ɪɚɞɢɚɧɵ, ɩɨɥɭɱɢɦ
18
π
=x
. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
10
0
=
sin18sin
−
=
1010
+
3
⋅
3
π
π
π
ɉɨɥɭɱɟɧɧɵɣ ɪɹɞ ɹɜɥɹɟɬɫɹ ɡɧɚɤɨɱɟɪɟɞɭɸɳɢɦɫɹ, ɱɥɟɧɵ ɤɨɬɨɪɨɝɨ ɭɛɵɜɚɸɬ ɩɨ
ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɢ ɫɬɪɟɦɹɬɫɹ ɤ ɧɭɥɸ. ɉɨɷɬɨɦɭ ɟɝɨ ɨɫɬɚɬɨɤ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ
3
ɩɟɪɜɨɝɨ ɨɬɛɪɨɲɟɧɧɨɝɨ ɱɥɟɧɚ. Ɍɚɤ ɤɚɤ
ɧɨɫɬɶɸ ɞɨ
0001,0
ɩɨɥɭɱɢɦ
π
10!3
⋅
0001,0
>
3
18sin
≈
−
=
1010
10!3
⋅
π
π
π
0
ȼɫɟ ɜɵɱɢɫɥɟɧɢɹ ɩɪɨɜɨɞɢɦ ɫ ɨɞɧɢɦ ɡɚɩɚɫɧɵɦ ɡɧɚɤɨɦ, ɩɨɥɚɝɚɹ
Ɍɚɤ ɤɚɤ
00624,313=π
, ɬɨ
12
−
n
xxx
⋅−
−
π
⋅
3
3
...
+
!12
5
10!510!3
, ɚ
.
...
−
.
5
5
π
10!5
⋅
0001,0
<
5
, ɬɨ ɫ ɬɨɱ-
.
14159,3≈π
.
107
18sin
x
x
x
10
14159,3
0
00624,31
6000
30899,000517,031416,0
=−=−=
.
ɂ ɬɚɤ,
3090,018sin0≈
. X
ɉɪɢɦɟɪ 7.3.
ȼɵɱɢɫɥɢɬɶ
ɫ ɬɨɱɧɨɫɬɶɸ
5ln
.
001,0
Ɋɟɲɟɧɢɟ.
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɹɞɨɦ ɢɡ ɩɪɢɦɟɪɚ 6.7:
−
1253
+
1
ln
−
1
§
x
¨
x
2
¨
x
©
53
n
xxx
++++⋅=
...
n
·
¸
.
+
...
¸
−
12
¹
1
Ⱦɪɨɛɶ
xx−+1
ɩɪɢ
ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɢɡ ɢɧɬɟɪɜɚɥɚ
()
1;1−∈x
∞+;0
, ɩɨ-
()
ɷɬɨɦɭ ɪɹɞ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɥɨɝɚɪɢɮɦɨɜ ɥɸɛɵɯ ɱɢɫɟɥ.
ɉɨ ɭɫɥɨɜɢɸ
x
1
+
, ɬɨɝɞɚ
5
=
1
−
,
551 −=+
,46 == xx
2
, ɩɨɥɭɱɚɟɦ
3
§·
ln5 2 ...
2 (0,6667 0,0988 0,0263 0,0084 0, 0029 0,0011 0,0004 ...).
≈⋅ ++++++++
21 2 12 12
=⋅ +⋅ +⋅ + ⋅ + ≈
¨¸
¨¸
333 5 3 7 3
©¹
357
§· §· §·
¨¸ ¨¸ ¨¸
©¹ ©¹ ©¹
Ɇɨɠɧɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɱɥɟɧɵ ɪɹɞɚ ɩɨɫɥɟ ɲɟɫɬɨɝɨ ɱɢɫɥɚ ɧɟ ɢɡɦɟɧɹɸɬ ɬɪɟɬɶɟɝɨ
ɱɢɫɥɚ ɩɨɫɥɟ ɡɚɩɹɬɨɣ ɜ ɡɧɚɱɟɧɢɢ ɫɭɦɦɵ, ɩɨɷɬɨɦɭ ɨɝɪɚɧɢɱɢɦɫɹ ɫɭɦɦɢɪɨɜɚɧɢɟɦ
ɜɟɥɢɱɢɧɵ ɲɟɫɬɢ ɫɥɚɝɚɟɦɵɯ ɢ ɩɨɥɭɱɢɦ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ
. X
608,15ln ≈
7.2. ɉɪɢɛɥɢɠɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɨɜ
ɉɪɢɛɥɢɠɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɢɧɬɟɝɪɚɥɨɜ ɨɫɧɨɜɚɧɨ ɧɚ ɪɚɡɥɨɠɟɧɢɢ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ.
ɉɪɢɦɟɪ 7.4.
ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ
Ɋɟɲɟɧɢɟ.
ɂɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɷɬɨɝɨ ɢɧɬɟɝɪɚɥɚ ɮɨɪɦɭɥɭ ɇɶɸɬɨɧɚ–
Ʌɟɣɛɧɢɰɚ ɦɵ ɧɟ ɦɨɠɟɦ, ɬɚɤ ɤɚɤ ɩɟɪɜɨɨɛɪɚɡɧɚɹ ɞɥɹ
3/1
2
x³−
ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
dxe
0
.
0001,0
2
x
−
ɯɨɬɹ ɢ ɫɭɳɟɫɬɜɭɟɬ, ɧɨ
e
108
ɧɟ ɜɵɪɚɠɚɟɬɫɹ ɜ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɹɯ (ɬɚɤɢɟ ɢɧɬɟɝɪɚɥɵ ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ
«ɧɟɛɟɪɭɳɢɦɢɫɹ»). ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɪɚɡɥɨɠɟɧɢɟɦ, ɩɨɥɭɱɟɧɧɵɦ ɜ ɩɪɢɦɟɪɟ 6.3:
2
−
x
e
1
642
xxx
...
+−+−=
.
!3!2!1
ɗɬɨɬ ɪɹɞ ɫɯɨɞɢɬɫɹ ɧɚ ɜɫɟɣ ɱɢɫɥɨɜɨɣ ɨɫɢ, ɡɧɚɱɢɬ, ɟɝɨ ɦɨɠɧɨ ɩɨɱɥɟɧɧɨ ɢɧɬɟɝɪɢɪɨɜɚɬɶ ɧɚ ɥɸɛɨɦ ɨɬɪɟɡɤɟ:
1/3 1/3
2
x
−
=−+−+⋅=
edx dx
³³
00
35 7
xx x
1/3
=− + − +=
x
0
⋅⋅ ⋅
31 52! 7 3!
11 1 1
=−+−+
3
⋅⋅ ⋅ ⋅ ⋅⋅
31!3 5 2!3 7 3!3
246
§·
xxx
1 ...
¨¸
1! 2 ! 3!
©¹
1/3 1/ 3 1/ 3
...
000
357
....
ɂɫɤɨɦɵɣ ɢɧɬɟɝɪɚɥ ɪɚɜɟɧ ɫɭɦɦɟ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ. Ɍɚɤ ɤɚɤ
1
5
2430
3!25
, ɚ
001,0
<=
1
1
3!13
⋅⋅
>=
3
81
,
001,0
ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɚɜɢɥɚ ɨɰɟɧɤɢ ɩɨɝɪɟɲɧɨɫɬɢ ɜ ɫɥɭɱɚɟ ɡɧɚ-
ɬɨ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
1
⋅⋅
001,0
ɤɨɱɟɪɟɞɭɸɳɟɝɨɫɹ ɪɹɞɚ, ɢɦɟɟɦ
3/1
2
−
x
dxe
³
0
1
1
−≈
3
3
33
⋅
. X
321,00123,03333,0
=−≈
ɉɪɢɦɟɪ 7.5.
1
sin
ȼɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ
x
ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ
dx
³
x
2/1
.
001,0
Ɋɟɲɟɧɢɟ.
Ʉɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɟɦ ɩɪɢɦɟɪɟ, ɢɧɬɟɝɪɚɥ ɹɜɥɹɟɬɫɹ «ɧɟɛɟɪɭɳɢɦɫɹ». ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɢɡɜɟɫɬɧɵɦ ɪɚɡɥɨɠɟɧɢɟɦ (6.11)
sin
x
1
x
42
xx
...
−+−=
.
!5!3
109
ɂɧɬɟɝɪɢɪɭɟɦ ɨɛɟ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɢɦ
1
sin
x
2/1
1
x
dx
§
¨
1
³³
¨
©
2/1
42
·
xx
¸
−+−=
!5!3
=⋅
dx
...
¸
¹
1
x=− + −=
1/ 2
33! 55!
⋅⋅
1/ 2 1/ 2
...
11
35
xx
=− − − + − −
¨¸
©¹
¨¸¨ ¸
233! 55!
⋅⋅
©¹© ¹
35
33!2 5512
⋅⋅ ⋅ ⋅
§·§ ·
11 1 1 1
§·
1....
ɉɨɥɭɱɟɧɧɵɣ ɪɹɞ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɚɡɧɨɫɬɶ ɞɜɭɯ ɫɯɨɞɹɳɢɯɫɹ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ ɪɹɞɨɜ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ Ʌɟɣɛɧɢɰɚ:
1
1 +
−
⋅
−
+
⋅
!771!551!33
⋅
1
1
ɢ
−
...
2
2!33
⋅⋅
1
+
2!55
⋅⋅
1
...
−
+
753
2!77
⋅⋅
Ɍɨɝɞɚ
1
sin
x
2/1
ª
x
1
1
−=³dx
«
⋅
¬
−
+
⋅
!771!551!33
⋅
ª
º
...
+
«
»
¼
¬
1
1
−−
2
2!33
⋅⋅
1
+
2!55
⋅⋅
1
−
⋅⋅
Ɍɚɤ ɤɚɤ ɜ ɡɧɚɤɨɱɟɪɟɞɭɸɳɟɦɫɹ ɪɹɞɟ, ɫɯɨɞɹɳɟɦɫɹ ɩɨ ɩɪɢɡɧɚɤɭ Ʌɟɣɛɧɢɰɚ, ɩɨ-
ɝɪɟɲɧɨɫɬɶ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ ɦɨɞɭɥɹ ɩɟɪɜɨɝɨ ɢɡ ɨɬɛɪɨɲɟɧɧɵɯ ɱɥɟɧɨɜ ɢ
1
<
!77
⋅
(ɞɥɹ ɩɟɪɜɨɝɨ ɪɹɞɚ),
001,0
1
5
2!55
⋅⋅
(ɞɥɹ ɜɬɨɪɨɝɨ ɪɹɞɚ),
001,0
<
ɬɨ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ ɢɦɟɟɦ
1
sin
³
x
2/1
ª
x
dx
1
1
«
¬
+
−=
⋅
ª
º
1
−−
«
»
2
!551!33
⋅
«
¼
¬
º
1
»
3
2!33
⋅⋅
»
¼
. X
453,0
≈
.
º
....
+
»
753
2!77
¼
110
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