Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Сходимость числовых рядов в примерах и задачах. Учебное пособие
.pdf
ɗɬɨɬɩɪɟɞɟɥɫɭɳɟɫɬɜɭɟɬɢɪɚɜɟɧ
ɚɡɧɚɱɢɬɫɭɦɦɚɢɫɯɨɞɧɨɝɨɪɹɞɚɪɚɜɧɚ
v
Ɉɬɜɟɬ
S
ɉɪɢɦɟɪɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
vS
f
¦
n
nn
Ɋɟɲɟɧɢɟɉɪɟɨɛɪɚɡɭɟɦɮɨɪɦɭɥɭɨɛɳɟɝɨɱɥɟɧɚɪɹɞɚ
f
f
f
nn
nn
f
¦¦
nn
f
§
¨
¨
©
nnnn
f
¦¦¦
nnn
n
nn
n
n
Ɋɚɫɫɦɨɬɪɢɦɱɚɫɬɢɱɧɵɟɫɭɦɦɵɷɬɨɝɨɪɹɞɚ
Sn
>@
nn
f
nnnnnn
f
·
¸¸
¦¦¦
¹
nnn
nnnn
§
¨
©
·
¸
nnnn
¹
>@
>@
>@
ɜɢɞɧɚɡɚɤɨɧɨɦɟɪɧɨɫɬɶɢɡɤɨɬɨɪɨɣɨɱɟɜɢɞɧɨɱɬɨ
>@
n
nnS
nS
nS
nnnn
nS
nn
ɉɨɨɩɪɟɞɟɥɟɧɢɸɫɭɦɦɵɪɹɞɚ
CBA
§
OLPOLP
SS
n
QQ
fofo
¨
©
·
¸
nn
¹
Ɍɨɝɞɚɫɭɦɦɚɢɫɯɨɞɧɨɝɨɪɹɞɚɪɚɜɧɚ
Ɉɬɜɟɬ
f
¦
nn
Q
Ɂɚɦɟɬɢɦɱɬɨɩɪɢɪɟɲɟɧɢɢ ɩɪɟɞɵɞɭɳɢɯ ɡɚɞɚɱ ɦɵ ɪɚɫɤɥɚɞɵɜɚɥɢɜɵɪɚɠɟ
ɧɢɟɩɨɞɫɭɦɦɨɣ ɧɚ ɞɪɨɛɢ ɩɪɟɨɛɪɚɡɨɜɵɜɚɹɜɵɪɚɠɟɧɢɟɨɛɳɟɝɨɱɥɟɧɚ ɪɹɞɚ Ⱦɥɹ
ɷɬɢɯ ɪɚɡɥɨɠɟɧɢɣ ɭɞɨɛɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫɬɚɧɞɚɪɬɧɵɣ ɫɩɨɫɨɛ ɩɨɥɭɱɟɧɢɹ ɫɭɦɦɵ
ɩɪɨɫɬɟɣɲɢɯ ɞɪɨɛɟɣ ɱɟɪɟɡ ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɋɚɫɫɦɨɬɪɢɦɟɝɨɧɚ
ɫɥɟɞɭɸɳɟɦɩɪɢɦɟɪɟ
ɉɪɢɦɟɪɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
f
¦
n
n
nnn
Ɋɟɲɟɧɢɟɇɚɯɨɞɢɬɶɫɭɦɦɭɪɹɞɚɛɭɞɟɦɩɨɨɩɪɟɞɟɥɟɧɢɸɬɨɟɫɬɶ
OLP
SSfo
n
n
Ɋɚɫɫɦɨɬɪɢɦɞɪɨɛɶɩɨɞɡɧɚɤɨɦɫɭɦɦɵɢɩɪɟɞɫɬɚɜɢɦɟɟɜɜɢɞɟɫɭɦɦɵɩɪɨ
ɫɬɟɣɲɢɯɞɪɨɛɟɣ
Ɍɚɤɤɚɤ
ɢɦɧɨɝɨɱɥɟɧ
ɢɦɟɟɬɤɨɪɧɢ
nn
n
nnn
ɇɚɣɞɟɦɧɟɢɡɜɟɫɬɧɵɟɤɨɷɮɮɢɰɢɟɧɬɵ
n
nnn
ɢ
n
n
nnn
nnnnnn
n
B
A
n
n
nnn
ɬɨ
C
n
nCnnBnnnA
ɬɨɝɞɚ
B
A
n
B
B
A
A
nnn
nCnn
n
ɉɪɢ
ɉɪɢ
ɉɪɢ
n
ɋɥɟɞɨɜɚɬɟɥɶɧɨ
f
n
ɉɨɥɭɱɢɦɮɨɪɦɭɥɭɞɥɹ
n ɨɣɱɚɫɬɢɱɧɨɣɫɭɦɦɵ
S
n
CCn
f
§
¨
¦¦
nnn
©
nn
S
S
·
¸
nnn
¹
S
SS
nnn
Ɂɧɚɱɢɬ
f
¦
n
n
nnn
n
OLPOLP
SS
n
n
§
¨
fofo
©
·
¸
nn
¹
Ɉɬɜɟɬ
f
ɉɪɢɦɟɪȾɨɤɚɡɚɬɶɱɬɨɪɹɞ
ɊɟɲɟɧɢɟɌɚɤɤɚɤɩɪɢɥɸɛɨɦ
¦
n
n
t
ɪɚɫɯɨɞɢɬɫɹ
ɜɵɩɨɥɧɹɟɬɫɹɧɟɪɚɜɟɧɫɬɜɨ
nk !
nk
ɬɨ
ɉɨɷɬɨɦɭɩɨɫɜɨɣɫɬɜɭɩɪɟɞɟɥɨɜ
ɢɫɥɟɞɨɜɚɬɟɥɶɧɨɪɹɞ
f
¦
n
Ɉɬɜɟɬɪɹɞɪɚɫɯɨɞɢɬɫɹ
n
S
¦
n
k
n
n
ɪɚɫɯɨɞɢɬɫɹ
n
n
k
nS
OLPOLP
fofo
n
n
t
n
f t
ɉɪɢɦɟɪɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
Ɋɟɲɟɧɢɟɉɪɟɨɛɪɚɡɭɟɦɜɢɞɪɹɞɚ
f
f
nn
nn
nn
f
¦¦
nn
f
f
¦
n
ɇɚɣɞɟɦ n ɭɸɱɚɫɬɢɱɧɭɸɫɭɦɦɭ
>@
>@
n
f
¦
n
nn
nS
«
nn
nn
nS
nn
nn
f
¦¦¦
nnn
nnnn
nn
nS
uSS
Ɍɨɝɞɚ
f
n
fofo
n
n
OLPOLP nS
Ɍɚɤ ɤɚɤ ɩɪɟɞɟɥ ɱɚɫɬɢɱɧɵɯ ɫɭɦɦ ɛɟɫɤɨɧɟɱɟɧ ɬɨ ɫɭɦɦɚ ɪɹɞɚ
f
¦
n
nn ɧɟɨɩɪɟɞɟɥɟɧɚɢɪɹɞ
f
¦
n
ɬɚɤɠɟɧɟɢɦɟɟɬ
nn
ɫɭɦɦɵ
Ɉɬɜɟɬɫɭɦɦɚɪɹɞɚɧɟɨɩɪɟɞɟɥɟɧɚ
f
ɉɪɢɦɟɪɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
¦
n
ɟɫɥɢ ɨɧɚ ɨɩɪɟɞɟ
nnn
ɥɟɧɚ
ɊɟɲɟɧɢɟɊɚɫɫɦɨɬɪɢɦɱɚɫɬɢɱɧɵɟɫɭɦɦɵɪɹɞɚ
Ɍɨɝɞɚ
n
n
OLP
ɉɪɟɞɟɥ
ɢɦɟɟɬɫɭɦɦɭ
Ɉɬɜɟɬ
±
ɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
>@
>@
nS
nS
>@
«
n
nnnn
ɱɚɫɬɢɱɧɨɣ ɫɭɦɦɵ ɤɨɧɟɱɟɧ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɪɹɞ ɫɯɨɞɢɬɫɹ ɢ
ɨɣn
SS
fonn
S
OLP
Ɂɚɞɚɱɢɞɥɹɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨɪɟɲɟɧɢɹ
nnS
fofofofo
nnn
OLP
fofo
nn
nS
OLPOLPOLPOLP nnnnS
nnnn
f
¦
n
nn
f
¦
n
nn
f
¦
n
nn
f
¦
n
n
nnn
f
¦
n
nn
f
¦
n
n
f
¦
n
Ⱦɨɤɚɡɚɬɶɱɬɨɪɹɞ
nn
n
ɂɫɫɥɟɞɨɜɚɬɶɪɹɞ
ɇɚɣɬɢɫɭɦɦɭɪɹɞɚ
f
¦
n
f
ɪɚɫɯɨɞɢɬɫɹ
¦
n
n
f
¦
n
f
¦
n
ɧɚɫɯɨɞɢɦɨɫɬɶ
nn
nnn
ɟɫɥɢɨɧɚɨɩɪɟɞɟɥɟɧɚ
nnnnn
ɇɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟɫɯɨɞɢɦɨɫɬɢɪɹɞɚ
Ɍɟɨɪɟɦɚɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ȿɫɥɢ ɪɹɞ
ɞɢɬɫɹɬɨɟɝɨ
ɬɨɟɫɬɶ
nɵɣɱɥɟɧɫɬɪɟɦɢɬɫɹ ɤ ɧɭɥɸ ɩɪɢɧɟɨɝɪɚɧɢɱɟɧɧɨɦɜɨɡɪɚɫɬɚɧɢɢn
u
OLP
n
fo
n
¦
ɗɬɨɭɫɥɨɜɢɟ ɧɟ ɹɜɥɹɟɬɫɹɞɨɫɬɚɬɨɱɧɵɦɞɥɹ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚɢɡɬɨɝɨ ɱɬɨ
OLP
u ɧɟ ɫɥɟɞɭɟɬ ɱɬɨɪɹɞɫɯɨɞɢɬɫɹ ɨɧ ɦɨɠɟɬ ɤɚɤɫɯɨɞɢɬɶɫɹ ɬɚɤ ɢ ɪɚɫɯɨ
fonn
ɞɢɬɶɫɹ
Ɍɟɨɪɟɦɚɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɪɚɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚȿɫɥɢnɣɱɥɟɧɪɹ
f
u ɧɟɫɬɪɟɦɢɬɫɹɤɧɭɥɸɩɪɢ fon ɬɨɷɬɨɬɪɹɞɪɚɫɯɨɞɢɬɫɹ
ɞɚ
¦
n
n
ɉɪɢɦɟɪ ɉɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨɝɨ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ
f
ɞɥɹɪɹɞɚ
¦
n
n
n
Ɋɟɲɟɧɢɟɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɩɪɢɡɧɚɤɚ
n
f
u
n
n
OLPOLP z
n
º
ª
»
«
f
¼
¬
OLP
nn
n
fofofo
n
ɇɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟɫɯɨɞɢɦɨɫɬɢɪɹɞɚɧɟɜɵɩɨɥɧɹɟɬɫɹɉɨɬɟɨɪɟɦɟɪɹɞ
ɪɚɫɯɨɞɢɬɫɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɧɟɜɵɩɨɥɧɹɟɬɫɹɪɹɞɪɚɫɯɨɞɢɬɫɹ
f
u ɫɯɨ
n
n
ɉɪɢɦɟɪ ɉɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨɝɨ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ
f
ɞɥɹɪɹɞɚ
¦
n
n
Ɋɟɲɟɧɢɟɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɭɫɥɨɜɢɹ
OLPOLP
u
n
n
n
fofo
n
ɇɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟɫɯɨɞɢɦɨɫɬɢɪɹɞɚɜɵɩɨɥɧɹɟɬɫɹɧɨɨɫɯɨɞɢɦɨɫɬɢɪɹ
ɞɚɧɢɱɟɝɨɫɤɚɡɚɬɶɧɟɥɶɡɹɞɚɥɟɟɛɭɞɟɬɞɨɤɚɡɚɧɨɱɬɨɞɚɧɧɵɣɪɹɞɪɚɫɯɨɞɢɬɫɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɜɵɩɨɥɧɹɟɬɫɹ
ɉɪɢɦɟɪɉɪɨɜɟɪɢɬɶɜɵɩɨɥɧɟɧɢɟ ɧɟɨɛɯɨɞɢɦɨɝɨ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ ɢ
ɫɞɟɥɚɬɶɜɵɜɨɞɨ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɟɫɥɢɷɬɨɜɨɡɦɨɠɧɨɞɥɹ ɪɹɞɚ ɢɦɟɸɳɟɝɨ ɱɚ
ɫɬɢɱɧɭɸɫɭɦɦɭ
n
S
n
n
Ɋɟɲɟɧɢɟ Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɧɟɨɛɯɨɞɢɦɨɝɨ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ ɧɚɞɨ ɡɧɚɬɶ
ɮɨɪɦɭɥɭɨɛɳɟɝɨɱɥɟɧɚɪɹɞɚɇɚɣɞɟɦɟɟ
n
SSu
nnn
n
n
n
n
n
n
nnn
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɧɟɨɛɯɨɞɢɦɨɝɨɩɪɢɡɧɚɤɚɫɯɨɞɢɦɨɫɬɢ
Ɍɚɤɤɚɤ
OLPOLP
u
n
fofo
n
u
n
fo
n
n
ɬɨɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɜɵɩɨɥɧɹɟɬɫɹ ɧɨ
OLP
nn
const
º
ª
«
¬
»
f
¼
ɨɫɯɨɞɢɦɨɫɬɢɪɹɞɚɧɢɱɟɝɨ ɫɤɚɡɚɬɶ ɧɟɥɶɡɹ ± ɨɧɦɨɠɟɬ ɤɚɤɫɯɨɞɢɬɶɫɹ ɬɚɤɢɪɚɫ
ɯɨɞɢɬɶɫɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɜɵɩɨɥɧɹɟɬɫɹ
ɉɪɢɦɟɪ Ⱦɥɹɪɹɞɚ
ɩɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɨɛɯɨ
ɞɢɦɨɝɨɭɫɥɨɜɢɹɫɯɨɞɢɦɨɫɬɢ ɢ ɫɞɟɥɚɬɶɜɵɜɨɞɨ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚɟɫɥɢɷɬɨɜɨɡ
ɦɨɠɧɨ
Ɋɟɲɟɧɢɟ Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɧɟɨɛɯɨɞɢɦɨɝɨ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ ɧɚɞɨ ɡɧɚɬɶ
ɮɨɪɦɭɥɭɨɛɳɟɝɨɱɥɟɧɚɪɹɞɚɇɚɣɞɟɦɟɟ
ɩɪɢ
un
ɩɪɢ
ɩɪɢ
un
un
ɢɬɞ
ȼɢɞɢɦ ɱɬɨ ɱɢɫɥɢɬɟɥɶ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɨɦɟɪɭ ɱɥɟɧɚ ɪɹɞɚ ɚ ɡɧɚɦɟɧɚɬɟɥɶ
ɩɪɟɞɫɬɚɜɥɹɟɬɫɨɛɨɣɫɬɟɩɟɧɶɞɜɨɣɤɢɬɨɟɫɬɶ
n
¦
n
n
ɉɪɨɜɟɪɢɦɜɵɩɨɥɧɟɧɢɟɧɟɨɛɯɨɞɢɦɨɝɨɩɪɢɡɧɚɤɚɫɯɨɞɢɦɨɫɬɢ
ª
u
n
n
n
«
n
fofo
¬
n
OLPOLP
fofofo
nn
ɱɟɦɛɵɫɬɪɟɟɬɫɹɭɜɟɥɢɱɢɜɚɸ
n
nn
ɡɧɚɱɟɧɢɹɩɪɢɧɨɢ
º
»
¼
Ɍɚɤɤɚɤ
fo
n
ɬɨɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɜɵɩɨɥɧɹɟɬɫɹ ɧɨ
u
OLP
n
ɨɫɯɨɞɢɦɨɫɬɢɪɹɞɚɧɢɱɟɝɨ ɫɤɚɡɚɬɶ ɧɟɥɶɡɹ ± ɨɧɦɨɠɟɬ ɤɚɤɫɯɨɞɢɬɶɫɹ ɬɚɤɢɪɚɫ
ɯɨɞɢɬɶɫɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɜɵɩɨɥɧɹɟɬɫɹ
ɉɪɢɦɟɪ Ⱦɥɹɪɹɞɚ
ɩɪɨɜɟɪɢɬɶ ɜɵɩɨɥɧɟɧɢɟ
ɧɟɨɛɯɨɞɢɦɨɝɨɭɫɥɨɜɢɹɫɯɨɞɢɦɨɫɬɢɢɫɞɟɥɚɬɶɜɵɜɨɞɨɫɯɨɞɢɦɨɫɬɢɪɹɞɚɟɫɥɢɷɬɨ
ɜɨɡɦɨɠɧɨ
Ɋɟɲɟɧɢɟɉɪɨɚɧɚɥɢɡɢɪɭɟɦɱɥɟɧɵɪɹɞɚɢɩɨɣɦɟɦɡɚɤɨɧɨɦɟɪɧɨɫɬɶ
un
un
un
ɢɬɞ
ȼɢɞɢɦɱɬɨɱɢɫɥɢɬɟɥɢɨɬɥɢɱɚɸɬɫɹɧɚɢɧɚɦɟɧɶɲɟɱɟɦɱɢɫɥɨɞɟɥɹɳɢ
ɟɫɹɧɚɬɨɟɫɬɶɢɦɟɸɬɜɢɞ
ɧɨɦɟɪɨɦ
nɜɬɨɪɨɣɦɧɨɠɢɬɟɥɶɜɫɟɝɞɚɧɟɱɟɬɧɵɣɢɞɥɹɧɟɝɨɫɩɪɚɜɟɞɥɢɜɨ
n
ɉɟɪɜɵɣ ɦɧɨɠɢɬɟɥɶ ɡɧɚɦɟɧɚɬɟɥɹɫɨɜɩɚɞɚɟɬ ɫ
n
n
Ɍɚɤɢɦɨɛɪɚɡɨɦ
u
n
nn
ɉɪɨɜɟɪɢɦɧɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟ
ª
«
n
n
nn
OLPOLP
u
n
OLP
n
«
nn
«
¬
n
fo
º
»
a
nn
»
»
a
nnn
¼
OLP
n
n
OLP
nnnn
fofofofofo
n
ɉɪɢ ɜɵɱɢɫɥɟɧɢɢ ɩɪɟɞɟɥɚ ɢɫɩɨɥɶɡɨɜɚɥɢ ɷɤɜɢɜɚɥɟɧɬɧɵɟ ɜɟɥɢɱɢɧɵ ɫɦ
ɩɪɢɥɨɠɟɧɢɟ
ɇɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɜɵɩɨɥɧɹɟɬɫɹ ɧɨ ɨ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
ɧɢɱɟɝɨɫɤɚɡɚɬɶɧɟɥɶɡɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɜɵɩɨɥɧɹɟɬɫɹ
ɉɪɢɦɟɪȾɥɹɪɹɞɚ
ɩɪɨɜɟɪɢɬɶɜɵɩɨɥɧɟɧɢɟɧɟɨɛɯɨɞɢ
ɦɨɝɨɭɫɥɨɜɢɹɫɯɨɞɢɦɨɫɬɢɢɫɞɟɥɚɬɶɜɵɜɨɞɨɫɯɨɞɢɦɨɫɬɢɪɹɞɚɟɫɥɢɷɬɨɜɨɡɦɨɠɧɨ
ɊɟɲɟɧɢɟɁɚɦɟɬɢɦɱɬɨɭɱɥɟɧɨɜɪɹɞɚɱɟɪɟɞɭɸɬɫɹɡɧɚɤɢɬɨɟɫɬɶɜɮɨɪɦɭ
n
ɥɟɨɛɳɟɝɨɱɥɟɧɚɪɹɞɚɞɨɥɠɟɧɩɪɢɫɭɬɫɬɜɨɜɚɬɶɦɧɨɠɢɬɟɥɶ
ɉɪɢ
ɩɪɢ
un
un ɬɞ
Ɂɧɚɱɟɧɢɟɱɢɫɥɢɬɟɥɟɣɜɪɚɡɚɛɨɥɶɲɟɧɨɦɟɪɚ
n
ɲɟɱɢɫɥɢɬɟɥɟɣɡɧɚɱɢɬ
u
n
n
n
nɚɡɧɚɦɟɧɚɬɟɥɢɧɚɛɨɥɶ
ɉɪɨɜɟɪɢɦɭɫɥɨɜɢɟɧɟɨɛɯɨɞɢɦɨɝɨɩɪɢɡɧɚɤɚ
n
n
§
¨
n
©
·
n
OLP
n
§
¸
¨
n
¹
©
n
·
n
OLP
¸
n
n
¹
ª
n
OLP
fofofo
«
¬
ɱɟɬɧɨɟɟɫɥɢ
n
ɧɟɱɟɬɧɨɟɟɫɥɢ
º
»
¼
ɉɪɟɞɟɥɧɟɫɭɳɟɫɬɜɭɟɬ ɬɚɤɤɚɤɟɝɨɡɧɚɱɟɧɢɟɧɟɨɩɪɟɞɟɥɟɧɨɋɥɟɞɨɜɚɬɟɥɶ
ɧɨɧɟɨɛɯɨɞɢɦɨɟɭɫɥɨɜɢɟɫɯɨɞɢɦɨɫɬɢɧɟɜɵɩɨɥɧɹɟɬɫɹɬɨɟɫɬɶɪɹɞɪɚɫɯɨɞɢɬɫɹ
Ɉɬɜɟɬɭɫɥɨɜɢɟɧɟɜɵɩɨɥɧɹɟɬɫɹɪɹɞɪɚɫɯɨɞɢɬɫɹ
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]