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Sonli qatorning asosiy tushunchalari. Qator
yaqinlashishining zaruriy shartlari. Yaqinlashuvchi
qatorlar va ularning xossalari. Garmonik qatorlar.
12-BOB. QATORLAR
§12.1Sonli qatorlar
4
0
4
)
4
2
(
1
n
x
dx
1
0
2
4
6
x
dx
0
2
1
dx
x
arctgx
0
2
4
x
dx
e
x
x
dx
1
0
2
)
(ln
1
)
1
ln(
dx
x
x
1
)
2
cos
1
(
dx
x
1
1
2
x
dx
1
0
3
2
2
1
cos
dx
x
x
0
3
)
1
(
x
dx
0
2
dx
e
x
2
0
)
6
2
(
2
cos
3
n
dx
x
3
1
2
)
6
2
(
1
n
dx
x
3
1
)
10
2
(
)
2
ln(
n
dx
x
1
1
2
)
4
(
1
n
x
dx
4
1
)
6
(
2
1
n
x
xdx
1.
Agar
...
...
3
2
1
n
u
u
u
u
qatorningbirinchi
n
tahadningyig’indisi
S
n
,
n
cheksizlikkaintilganda
n
chekli
S
limitgaintilsa:
S
S
n
n
lim
,
qatoryaqinlashuvchideyiladi.
S
sonyaqinlashuvchiqatorningyig’indisideyiladi.
Qatorningyaqinlashuvchibo’lishiuchun
n
cheksizlikkaintilganda
n
u
n
ningnolgaintilishi
0
n
u
zarurdir (ammoyetarliemas).
2.
Hadlarimusbathamdakamayuvchibo’lganqatorningyaqinlashishiuchunintegralalomat:
Agar
n
f
u
n
deb olinsa, bunda
n
f
kamayuvchi funksiya va
1
.
,
'
,
,
'
i
uzoqlashad
qator
holda
u
lsa
bo
di
yaqinlasha
qator
holda
u
lsa
bo
A
dx
x
f
3
.
Musbat hadli qatorning yaqinlashishi uchun Dalamber alomati:
Agar
.
,
'
1
,
,
'
1
,
,
'
1
lim
1
qoladi
yechilmay
masala
holda
u
lsa
bo
i
uzoqlashad
qator
holda
u
lsa
bo
di
yaqinlasha
qator
holda
u
lsa
bo
r
u
u
D
n
n
n
4.
Musbat hadli qatorlarning yaqinlashishi uchun Koshi alomati
.
1
,
1
,
1
lim
aydi
tekshirilm
bilan
alomat
bu
K
chi
uzoqlashuv
qator
K
vchi
yaqinlashu
qator
K
u
K
n
n
n
5.
Musbathadliqatorlarnitaqoslash
...
...
3
2
1
n
u
u
u
u
(12.1)
...
...
3
2
1
n
v
v
v
v
(12.2)
ikkitamusbathadliqatorbo’lsin.
1) Agar
n
n
v
u
bo’lib, (12.2) qatoryaqinlashsa, uholda (12.1) qatorhamyaqinlashadi.
2) Agar
n
n
v
u
bo’lib, (12.1) qatoruzoqlashsa, uholda (12.2) qatorhamuzoqlashadi.
6.
Agarishoralarinavbatlashuvchi
...
4
3
2
1
u
u
u
u
qatorda
...
3
2
1
u
u
u
va
0
lim
n
n
u
bo’lsa,
qatoryaqinlashuvchibo’ladi.
7.
Absolyut yaqinlashish
...
...
3
2
1
n
u
u
u
u
(12.3)
qator hadlarning absolyut qiymatlaridan tuzilgan
...
...
3
2
1
n
u
u
u
u
(12.4)
qator yaqinlashsa, (12.3) qator ham yaqinlashadi. Bu holda (12.3) qator yaqinlashuvchi bo’lib (12.4) qator
uzoqlashuvchi bo’lsa, (12.3) qator absolyut yaqinlashuvchi deyiladi. (12.3) qator shartli (absolyutmas) yaqinlashuvchi
deyiladi.
“A” guruh
Quyidagi qatorlar uchun yaqinlashishning zaruriy sharti bajariladimi?
12.1.
...
8
7
6
5
4
3
2
1
12.2.
...
7
1
5
1
3
1
1
1
12.3.
...
81
8
27
6
9
4
3
2
Qatorlar uchun: 1) qatorni yaqinlashishini isbotlang; 2) qatorning yig’indisi (S) ni toping.
12.4)
∙
∙
. . .
. ..
12.5)
∙
∙
. . .
. ..
12.6)
∙
∙
. . .
. ..
12.7)
∙
∙
. . .
. ..
12.8)
∙
∙
. . .
. ..
12.9)
∙ ∙
∙ ∙
. . .
. ..
12.10)
. . .
. ..
12.11)
. . .
. ..
12.12)
. . .
. ..
12.13)
𝑎𝑟𝑐𝑡𝑔
𝑎𝑟𝑐𝑡𝑔
. . . 𝑎𝑟𝑐𝑡𝑔
. ..
12.14)
...
16
3
8
3
4
3
2
3
12.15)
1
3
2
1
n
n
n
12.16).
1
2
2
2
3
1
3
1
6
n
n
n
n
12.17)
1
2
5
3
5
ln
n
n
n
.
Qatorlarning yaqinlashishini solishtirish alomati yordamida yeching.
12.18)
...
2
1
2
1
...
2
3
1
2
1
1
1
2
3
n
n
12.19)
...
2
sin
...
4
sin
2
sin
n
12.20)
...
1
1
...
2
1
2
1
1
2
2
n
n
12.21)
...
4
1
1
...
6
3
1
5
2
1
n
n
12.22)
...
2
1
...
8
3
3
2
n
n
n
12.23)
...
4
...
8
4
n
tg
tg
tg
12.24)
...
1
1
...
5
1
2
1
2
n
12.25)
...
1
3
1
...
5
1
2
1
n
12.26)
...
1
ln
1
...
3
ln
1
2
ln
1
n
12.27)
1
2
5
4
1
n
n
n
12.28)
1
2
3
2
1
1
n
n
n
12.29)
1
2
2
1
n
n
n
12.30)
1
1
n
n
n
12.31)
1
4
1
1
n
n
12.32.
1
2
5
3
2
n
n
n
12.33.
1
3
11
3
7
2
n
n
n
12.34.
1
5
2
12
7
n
n
n
12.35.
1
3
ln
1
n
n
12.36.
1
4
3
3
n
n
n
n
12.37.
1
2
3
ln
n
n
n
12.38.
1
3
1
ln
1
n
n
n
12.39.
1
5
3
1
ln
1
n
n
n
12.40.
12.41.
1
2
1
cos
1
n
n
n
Musbat hadli qatorlarni taqqoslash teoremalari.
Musbat hadli sonli qatorlar yaqinlashishining etarli
shartlari: Dalamber alomati, Koshining radikal va
integral alomatlari.
Qatorlarning yaqinlashishini Dalamber alomati yordamida isbotlang.
12.42)
...
!
1
2
1
...
!
5
1
!
3
1
n
12.43)
...
2
...
2
2
2
1
2
n
n
12.44)
...
2
...
8
2
4
1
n
ntg
tg
tg
12.45)
...
3
4
...
5
1
1
3
...
5
2
...
5
1
5
2
1
2
n
n
12.46)
...
3
...
9
4
3
1
2
n
n
12.47)
...
!
3
1
2
...
3
1
...
6
3
3
1
3
1
n
n
n
12.48)
...
2
sin
...
4
sin
4
2
sin
2
n
n
12.49)
...
!
1
...
!
3
2
!
2
1
n
n
12.50)
...
!
2
!
1
...
4
2
3
2
2
2
n
n
n
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