Bài tập toán cao cấp tập 3 part 6 ppsx

12.4. T´ıch phˆan m˘a
.
t 165
V´ı du
.
4. T´ınh t´ıch phˆan

(σ)
2dxdy + ydxdz −x
2
zdydz, trong d´o(σ)
l`a ph´ıa trˆen cu
’
a phˆa
`
n elipxoid 4x
2
+ y
2
+4z
2
=1n˘a
`
m trong g´oc phˆa
`
n
t´am I.
Gia
’
i. Ta viˆe
´

tt´ıch phˆan d
˜a cho du
.
´o
.
ida
.
ng
I =2

(σ)
dxdy +

(σ)
ydydz −

(σ)
x
2
zdydz.
v`a su
.
’
du
.
ng phu
.
o
.
ng tr`ınh cu

’
am˘a
.
t(σ)d
ˆe
’
biˆe
´
ndˆo
’
imˆo
˜
i t´ıch phˆan. Lu
.
u
´yr˘a
`
ng cos α>0, cos β>0, cos γ>0.
(i) V`ı h`ınh chiˆe
´
ucu
’
am˘a
.
t(σ)lˆen m˘a
.
t ph˘a
’
ng Oxy l`a phˆa
`

ntu
.
h`ınh
elip
x
2
1
2
+
y
2
2
2
 1nˆen
I
1
=

(σ)
dxdy =

D(x,y)
dxdy =
π
2
(v`ı diˆe
.
n t´ıch elip = 2π)
(ii) H`ınh chiˆe
´

ucu
’
a(σ)lˆen m˘a
.
t ph˘a
’
ng Oxz l`a phˆa
`
ntu
.
h`ınh tr`on
4x
2
+4z
2
 4 ⇔ x
2
+ z
2
 1. M˘a
.
t kh´ac t `u
.
phu
.
o
.
ng tr`ınh m˘a
.
tr´ut ra

y =2

1 − x
2
− y
2
v`a do d´o
I
2
=

(σ)
ydxdz =2

D(x,y)
√
1 − x
2
− z
2
dxdz = |chuyˆe
’
n sang to
.
adˆo
.
cu
.
.
c|

=2
π/2

0
dϕ
1

0
√
1 −r
2
rdr =
π
3
·
(iii) H`ınh chiˆe
´
ucu
’
a(σ)lˆen m˘a
.
t ph˘a
’
ng Oyz l`a mˆo
.
t phˆa
`
ntu
.
h`ınh

elip
y
2
4
+ z
2
 1(y  0, z  0). T`u
.
phu
.
o
.
ng tr`ınh m˘a
.
t(σ)r´ut ra
166 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
x =

1 −
y
2
4

− z
2
rˆo
`
ithˆe
´
v`ao h`am du
.
´o
.
idˆa
´
u t´ıch phˆan cu
’
a I
3
:
I
3
=

(σ)
x
2
zdydz =

D(y,z)
z

1 −

y
2
4
− z
2

dydz
=
1

0
dz
2
√
1−z
2

0
z

1 −
y
2
4
−z
2

dy = ···=
4
15

·
Nhu
.
vˆa
.
y I =2I
1
+ I
2
− I
3
=
4π
3
−
4
15
· 
V´ı d u
.
5. T´ınh

(σ)
−
ydydz, trong d´o(σ) l`a m˘a
.
tcu
’
at´u
.

diˆe
.
n gi´o
.
iha
.
n
bo
.
’
im˘a
.
t ph˘a
’
ng x +y +z = 1 v`a c´ac m˘a
.
t ph˘a
’
ng to
.
ad
ˆo
.
, t´ıch phˆan du
.
o
.
.
c
lˆa

´
y theo ph´ıa trong cu
’
at´u
.
diˆe
.
n.
Gia
’
i. M˘a
.
t ph˘a
’
ng x + y + z =1c˘a
´
t c´ac tru
.
cto
.
ad
ˆo
.
ta
.
i A(1, 0, 0),
B(0, 1,0) v`a C =(0, 0, 1). Ta k´y hiˆe
.
ugˆo
´

cto
.
ad
ˆo
.
l`a O(0, 0,0). T`u
.
d
´o
suy ra m˘a
.
tk´ın (σ)gˆo
`
mt`u
.
4 h`ınh tam gi´ac ∆ABC,∆BCO,∆ACO
v`a ∆ABO. Do vˆa
.
y t´ıch phˆan d
˜a cho l`a tˆo
’
ng cu
’
abˆo
´
n t´ıch phˆan.
(i) T´ıch phˆan I
1
=


ABC
ydxdz.R´ut y t`u
.
phu
.
o
.
ng tr`ınh m˘a
.
t(σ) ⊃
∆ABC ta c´o y =1−x −z v`a do d
´o
I
1
= −

ACO
(1 − x −z)dxdz =
1

0
dx
1−x

0
(x + z −1)dz = −
1
6
·
(Lu

.
u´yr˘a
`
ng cos β = cos(n , O y ) < 0 v`ı vecto
.
n lˆa
.
pv´o
.
ihu
.
´o
.
ng du
.
o
.
ng
tru
.
c Oy mˆo
.
t g´oc t`u, do d
´o tru
.
´o
.
c t´ıch phˆan theo ∆ACO xuˆa
´
thiˆe

.
ndˆa
´
u
tr `u
.
)
(ii)

(BCD)
ydxdz =

(ABO)
ydxdz =0
12.4. T´ıch phˆan m˘a
.
t 167
v`ım˘a
.
t ph˘a
’
ng BCO v`a ABO dˆe
`
u vuˆong g´oc v´o
.
im˘a
.
t ph˘a
’
ng Oxz.

(iii)

(ACO)
ydxdz =

ACO
0dxdz =0.
Vˆa
.
y I = −
1
6
. 
V´ı du
.
6. T´ınh t´ıch phˆan I =

(σ)
x
3
dydz + y
3
dzdx + z
3
dxdy, trong
d
´o ( σ) l`a ph´ıa ngo`ai m˘a
.
tcˆa
`

u x
2
+ y
2
+ z
2
= R
2
.
Gia
’
i.
´
Ap du
.
ng cˆong th´u
.
c Gauss-Ostrogradski ta c´o

(σ)
=3

D
(x
2
+ y
2
+ z
2
)dxdydz

trong d
´o D ⊂ R
3
l`a miˆe
`
nv´o
.
i biˆen l`a m˘a
.
t(σ). Chuyˆe
’
n sang to
.
ad
ˆo
.
cˆa
`
u ta c´o
3

D
(x
2
+ y
2
+ z
2
)dxdydz =3
2π


0
dϕ
π

0
sin θdθ
R

0
r
4
dr
=
12πR
5
5
·
Vˆa
.
y I =
12πR
5
5
· 
V´ı du
.
7. T´ınh t´ıch phˆan

L

x
2
y
3
dx + dy + zdz, trong d´o L l`a du
.
`o
.
ng
tr`on x
2
+ y
2
=1,z = 0, c`on m˘a
.
t(σ) l`a ph´ıa ngo`ai cu
’
anu
.
’
am˘a
.
tcˆa
`
u
x
2
+ y
2
+ z

2
=1,z>0v`aL c´o di
.
nh hu
.
´o
.
ng du
.
o
.
ng.
Gia
’
i. Trong tru
.
`o
.
ng ho
.
.
p n`ay P = x
2
y
3
, Q =1,R = z.Dod´o
∂Q
∂x
−
∂P

∂y
= −3x
2
y
2
,
∂R
∂y
−
∂Q
∂z
=0,
∂P
∂z
−
∂R
∂x
=0
168 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
v`a do d´o theo cˆong th´u
.
c Stokes ta c´o


L
= −3

(σ)
x
2
y
2
dxdy = −
π
8
· 
B
`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan m˘a
.
t theo diˆe
.
n t´ıch sau d
ˆay
1.

(Σ)
(x + y + z)dS, (Σ) l`a m˘a

.
tlˆa
.
pphu
.
o
.
ng 0  x  1, 0  1,
0  z  1. (D
S. 9)
2.

(Σ)
(2x+y +z)dS, (Σ) l`a phˆa
`
nm˘a
.
t ph˘a
’
ng x+y +z =1n˘a
`
m trong
g´oc phˆa
`
n t´am I.(D
S.
2
√
3
3

)
3.

(Σ)

z +2x +
4y
3

dS, (Σ) l`a phˆa
`
nm˘a
.
t ph˘a
’
ng 6x +4y +3z =12
n˘a
`
m trong g´oc phˆa
`
n t´am I. (D
S. 4
√
61)
4.

(σ)

x
2

+ y
2
dS, (Σ) l`a phˆa
`
nm˘a
.
t n´on z
2
= x
2
+ y
2
,0 z  1.
(D
S.
2
√
2π
3
)
5.

(Σ)
(y + z +
√
a
2
− x
2
)dS, (Σ) l`a phˆa

`
nm˘a
.
t tru
.
x
2
+ y
2
= a
2
n˘a
`
m
gi˜u
.
a hai m˘a
.
t ph˘a
’
ng z =0v`az = h.(D
S. ah(4a + πh))
6.

(Σ)

y
2
− x
2

dS, (Σ) l`a phˆa
`
nm˘a
.
t n´on z
2
= x
2
+ y
2
n˘a
`
m trong m˘a
.
t
tru
.
x
2
+ y
2
= a
2
.(DS.
8a
3
3
)
12.4. T´ıch phˆan m˘a
.

t 169
7.

(Σ)
(x + y + z)dS, (Σ) l`a nu
.
’
a trˆen cu
’
am˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= a
2
.
(D
S. πa
3
)
8.

(Σ)


x
2
+ y
2
dS, (Σ) l`a m˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= a
2
.(DS.
8πa
3
3
)
9.

(Σ)
dS
(1 + x + y)
, (Σ) l`a biˆen cu
’
at´u
.

diˆe
.
n x´ac d
i
.
nh bo
.
’
ibˆa
´
tphu
.
o
.
ng
tr`ınh x+y+z  1, x  0, y  0, z  0. (D
S.
1
3
(3−
√
3) +(
√
3−1) ln 2)
10.

(Σ)
(x
2
+ y

2
)dS, (Σ) l`a phˆa
`
nm˘a
.
t paraboloid x
2
+ y
2
=2z du
.
o
.
.
c
c˘a
´
trabo
.
’
im˘a
.
t ph˘a
’
ng z = 1. (D
S.
55 + 9
√
3
65

)
11.

(Σ)

1+4x
2
+4y
2
dS, (Σ) l`a phˆa
`
nm˘a
.
t paraboloid z =1−x
2
−y
2
gi´o
.
iha
.
nbo
.
’
i c´ac m˘a
.
t ph˘a
’
ng z =0v`az = 1. (D
S. 3π)

12.

(Σ)
(x
2
+ y
2
)dS, (Σ) l`a phˆa
`
nm˘a
.
t n´on z =

x
2
+ y
2
n˘a
`
mgi˜u
.
a
c´ac m˘a
.
t ph˘a
’
ng z =0v`az = 1. (D
S.
π
√

2
2
)
13.

(Σ)
(xy + yz + zx)dS, (Σ) l`a phˆa
`
nm˘a
.
t n´on z =

x
2
+ y
2
n˘a
`
m
trong m˘a
.
t tru
.
x
2
+ y
2
=2ax (a>0). (DS.
64a
4

√
2
15
)
14.

(Σ)
(x
2
+ y
2
+ z
2
)dS, (Σ) l`a ma
.
tcˆa
`
u. (DS. 4π)
15.

(Σ)
xds, (Σ) l`a phˆa
`
nm˘a
.
tdu
.
o
.
.

cc˘a
´
trat`u
.
partab oloid 10x = y
2
+z
2
170 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
bo
.
’
im˘a
.
t ph˘a
’
ng x = 10. (D
S.
50π
3
(1 + 25
√

5))
Su
.
’
du
.
ng cˆong th´u
.
c t´ınh diˆe
.
n t´ıch m˘a
.
t S(Σ) =

(Σ)
dS dˆe
’
t´ınh diˆe
.
n
t´ıch cu
’
a phˆa
`
nm˘a
.
t (Σ) nˆe
´
u
16. (Σ) l`a phˆa

`
nm˘a
.
t ph˘a
’
ng 2x +2y + z =8a n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
= R
2
.(DS. 3πR
2
)
17. (Σ) l`a phˆa
`
nm˘a
.
t tru
.
y + z
2
= R
2

n˘a
`
m trong m˘a
.
t tru
.
x
2
+ y
2
= R
2
.(DS. 8R
2
)
18. (Σ) l`a phˆa
`
nm˘a
.
t paraboloid x
2
+ y
2
=6z n˘a
`
m trong m˘a
.
t tru
.
x

2
+ y
2
= 27. (DS. 42π)
19. (Σ) l`a phˆa
`
nm˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
=3a
2
n˘a
`
m trong paraboloid
x
2
+ y
2
=2az.(DS. 2πa
2
(3 −
√
3))

20. (Σ) l`a phˆa
`
nm˘a
.
t n´on z
2
=2xy n˘a
`
m trong g´oc phˆa
`
n t´am I gi˜u
.
a
hai m˘a
.
t ph˘a
’
ng x =2,y = 4. (D
S. 16)
21. (Σ) l`a phˆa
`
nm˘a
.
t tru
.
x
2
+ y
2
= Rx n˘a

`
m trong m˘a
.
tcˆa
`
u
x
2
+ y
2
+ z
2
= R
2
.(DS. 4R
2
)
T´ınh c´ac t´ıch phˆan m˘a
.
t theo to
.
ad
ˆo
.
sau:
22.

(Σ)
dxdy, (Σ) l`a ph´ıa ngo`ai phˆa
`

nm˘a
.
t n´on z =

x
2
+ y
2
khi
0  z  1. (D
S. −π)
23.

(Σ)
ydzdx, (Σ) l`a ph´ıa trˆen cu
’
a phˆa
`
nm˘a
.
t ph˘a
’
ng x + y + z = a
(a>0) n˘a
`
m trong g´oc phˆa
`
n t´am I.(D
S.
a

3
6
)
24.

(Σ)
xdydz + ydzdx+ zdxdy, (Σ) l`a ph´ıa trˆen cu
’
a phˆa
`
nm˘a
.
t ph˘a
’
ng
x + z −1=0n˘a
`
mgi˜u
.
a hai m˘a
.
t ph˘a
’
ng y =0v`ay = 4 v`a thuˆo
.
c v`ao
g´oc phˆa
`
n t´am I. (D
S. 4)

12.4. T´ıch phˆan m˘a
.
t 171
25.

(Σ)
− xdydz + zdzdx +5dxdy, (Σ) l`a ph´ıa trˆen cu
’
a phˆa
`
nm˘a
.
t
ph˘a
’
ng 2x +3y + z = 6 thuˆo
.
c g´oc phˆa
`
n t´am I. (D
S. 6)
26.

(Σ)
yzdydz + xzdxdz + xydxdy, (Σ) l`a ph´ıa trˆen cu
’
a tam gi´ac ta
.
o
bo

.
’
i giao tuyˆe
´
ncu
’
am˘a
.
t ph˘a
’
ng x + y + z = a v´o
.
i c´ac m˘a
.
t ph˘a
’
ng to
.
a
d
ˆo
.
.(DS.
a
4
8
)
27.

(Σ)

x
2
dydz + z
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
a phˆa
`
nm˘a
.
t n´on
x
2
+ y
2
= z
2
,0 z  1. (DS. −
4
3
)
28.

(Σ)
xdydz + ydzdx + zdxdy, (Σ) l`a ph´ıa ngo`ai phˆa
`
nm˘a
.
tcˆa
`

u
x
2
+ y
2
+ z
2
= a
2
.(DS. 4πa
3
)
29.

(σ)
x
2
dydz −y
2
dzdx + z
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
am˘a
.
tcˆa
`
u
x
2

+ y
2
+ z
2
= R
2
thuˆo
.
c g´oc phˆa
`
n t´am I. (DS.
πa
4
8
)
30.

(Σ)
2dxdy + ydzdx − x
2
zdydz, (Σ) l`a ph´ıa ngo`ai cu
’
a phˆa
`
nm˘a
.
t
elipxoid 4x
2
+ y

2
+4z
2
= 4 thuˆo
.
c g´oc phˆa
`
n t´am I. (DS.
4π
3
−
4
15
)
31.

(Σ)
(y
2
+ z
2
)dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
am˘a
.
t tru
.
z
2
=1− x

2
,
0  y  1. (D
S.
π
3
)
32.

(Σ)
(z −R)
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
anu
.
’
am˘a
.
tcˆa
`
u
x
2
+ y
2
+(z −R)
2
= R
2

, R  z  2R.(DS. −
5π
24
)
172 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
33.

(Σ)
x
2
dydz + y
2
dzdx + z
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
a phˆa
`
nm˘a
.
t
cˆa

`
u x
2
+ y
2
+ z
2
= a
2
thuˆo
.
c g´oc phˆa
`
n t´am I. (DS.
3πa
4
8
)
34.

(Σ)
z
2
dxdy,(σ) l`a ph´ıa trong cu
’
am˘a
.
t elipxoid
x
2

+ y
2
+2z
2
= 2. (DS. 0)
35.

(Σ)
(z +1)dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
am˘a
.
tcˆa
`
u
x
2
+ y
2
+ z
2
= R
2
.(DS.
4πR
3
3
)
36.


(Σ)
x
2
dydz + y
2
dzdx + z
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
am˘a
.
tcˆa
`
u
(x −a)
2
+(y − b)
2
+(z − c)
2
= R
2
.(DS.
8πR
3
3
(a + b + c))
37.

(Σ)

x
2
y
2
zdxdy, (Σ) l`a ph´ıa trong cu
’
anu
.
’
adu
.
´o
.
im˘a
.
tcˆa
`
u
x
2
+ y
2
+ z
2
= R
2
.(DS.
2πR
7
105

)
38.

(Σ)
xzdxdy + xydydz + yzdxdz, (Σ) l`a ph´ıa ngo`ai cu
’
at´u
.
diˆe
.
nta
.
o
bo
.
’
i c´ac m˘a
.
t ph˘a
’
ng to
.
ad
ˆo
.
v`a m˘a
.
t ph˘a
’
ng x + y + z = 1. (DS.

1
8
)
Chı
’
dˆa
˜
n. Su
.
’
du
.
ng nhˆa
.
n x´et nˆeu trong phˆa
`
nl´ythuyˆe
´
t.
39.

(Σ)
yzdydz + xzdxdz + xydxdy, (Σ) l`a ph´ıa ngo`ai cu
’
am˘a
.
tbiˆen
t´u
.
diˆe

.
nlˆa
.
pbo
.
’
i c´ac m˘a
.
t ph˘a
’
ng x =0,y =0,z =0,x + y + z = a.
(D
S. 0)
40.

(Σ)
x
2
dydz + y
2
dzdx + z
2
dxdy, (Σ) l`a ph´ıa ngo`ai cu
’
anu
.
’
a trˆen
12.4. T´ıch phˆan m˘a
.

t 173
m˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= R
2
(z  0). (DS.
πR
4
2
)
´
Ap du
.
ng cˆong th´u
.
c Gauss-Ostrogradski d
ˆe
’
t´ınh t´ıch phˆan m˘a
.
t theo
ph´ıa ngo`ai cu

’
am˘a
.
t (Σ) (nˆe
´
um˘a
.
t khˆong k´ın th`ı bˆo
’
sung d
ˆe
’
n´o tro
.
’
th`anh
k´ın)
41.

(Σ)
x
2
dydz + y
2
dzdx + z
2
dxdy, (Σ) l`a m˘a
.
tcˆa
`

u
(x −a)
2
+(y −b)
2
+(z −c)
2
= R
2
.(DS.
8π
3
(a + b + c)R
3
)
42.

(Σ)
xdydz + ydzdx + zdxdy, (Σ) l`a m˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= R
2

.
(D
S. 4πR
3
)
43.

(Σ)
4x
3
dydz +4y
3
dzdx − 6z
2
dxdy, (Σ) l`a biˆen cu
’
a phˆa
`
n h`ınh
tru
.
x
2
+ y
2
 a
2
,0 z  h.(DS. 6πa
2
(a

2
− h
2
))
44.

(σ)
(y −z)dydz +(z −x)dzdx +(x − y)dxdy, (Σ) l`a phˆa
`
nm˘a
.
t
n´on x
2
+ y
2
= z
2
,0 x  h.(DS. 0)
Chı
’
dˆa
˜
n. V`ı (Σ) khˆong k´ın nˆen cˆa
`
nbˆo
’
sung phˆa
`
nm˘a

.
t ph˘a
’
ng z = h
n˘a
`
m trong n´on d
ˆe
’
thu du
.
o
.
.
cm˘a
.
tk´ın.
45.

(Σ)
dydz + zxdzdx + xydxdy, (Σ) l`a biˆen cu
’
amiˆe
`
n
{(x, y, z):x
2
+ y
2
 a

2
, 0  z  h}.(DS. 0)
46.

(Σ)
ydydz + zdzdx + xdxdy, (Σ) l`a m˘a
.
tcu
’
a h`ınh ch´op gi´o
.
iha
.
n
bo
.
’
i c´ac m˘a
.
t ph˘a
’
ng
x + y + z = a (a>0), x =0,y =0,z = 0. (D
S. 0)
47.

(Σ)
x
3
dydz + y

3
dzdx + z
3
dxdy, (Σ) l`a m˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= x.
174 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
(DS.
π
5
)
48.

(Σ)

x
3
dydz + y
3
dzdx + z
3
dxdy, (Σ) l`a m˘a
.
tcˆa
`
u x
2
+ y
2
+ z
2
= a
2
.
(D
S.
12πa
5
5
)
49.

(Σ)
z
2

dxdy, (Σ) l`a m˘a
.
t elipxoid
x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1. (DS. 0)
Chı
’
dˆa
˜
n. Xem v´ıdu
.
10, mu
.
cIII.
50.

(Σ)

xdydz +ydzdx+zdxdy, (Σ) l`a m˘a
.
t elipxoid
x
2
a
2
+
y
2
b
2
+
z
2
c
2
=1.
(D
s. 4πabc)
51.

(Σ)
xdydz + ydzdx + zdxdy, (Σ) l`a biˆen h`ınh tru
.
x
2
+ y
2
 a

2
,
−h  z  h.(D
S. 6πa
2
h)
52.

(Σ)
x
2
dydz + y
2
dzdx + z
2
dxdy, (Σ) l`a biˆen cu
’
a h`ınh lˆa
.
pphu
.
o
.
ng
0  x  a,0 y  a,0 z  a.(D
S. 3a
4
)
D
ˆe

’
´ap du
.
ng cˆong th´u
.
c Stokes, ta lu
.
u´yla
.
iquyu
.
´o
.
c
Hu
.
´o
.
ng du
.
o
.
ng cu
’
a chu tuyˆe
´
n ∂Σcu
’
am˘a
.

t (Σ) d
u
.
o
.
.
cquyu
.
´o
.
cnhu
.
sau: Nˆe
´
umˆo
.
t ngu
.
`o
.
i quan tr˘a
´
cd
´u
.
ng trˆen ph´ıa d
u
.
o
.

.
ccho
.
ncu
’
am˘a
.
t(t´u
.
c
l`a hu
.
´o
.
ng t`u
.
chˆan d
ˆe
´
ndˆa
`
utr`ung v´o
.
ihu
.
´o
.
ng cu
’
a vecto

.
ph´ap tuyˆe
´
n) th`ı
khi ngu
.
`o
.
i quan s´at di chuyˆe
’
n trˆen ∂Σ theo hu
.
´o
.
ng d
´o th`ı m˘a
.
t (Σ) luˆon
luˆon n˘a
`
m bˆen tr´ai.
´
Ap du
.
ng cˆong th´u
.
c Stokes d
ˆe
’
t´ınh c´ac t´ıch phˆan sau

53.

C
xydx + yzdy + xzdz, C l`a giao tuyˆe
´
ncu
’
am˘a
.
t ph˘a
’
ng 2x −3y +
4z − 12 = 0 v´o
.
i c´ac m˘a
.
t ph˘a
’
ng to
.
ad
ˆo
.
.(DS. −7)
12.4. T´ıch phˆan m˘a
.
t 175
54.

C

ydx+zdy+xdz, C l`a du
.
`o
.
ng tr`on x
2
+y
2
+z
2
= R
2
, x+y+z =0
c´o hu
.
´o
.
ng ngu
.
o
.
.
cchiˆe
`
u kim d
ˆo
`
ng hˆo
`
nˆe

´
u nh`ın t`u
.
phˆa
`
ndu
.
o
.
ng tru
.
c Ox.
(D
S. −
√
3πR
2
)
55.

C
(y − z)dx +(z − x)dy +(x − y)dz, C l`a elip x
2
+ y
2
= a
2
,
x
a

+
z
h
=1(a>0, h>0) c´o hu
.
´o
.
ng ngu
.
o
.
.
c chiˆe
`
u kim d
ˆo
`
ng hˆo
`
nˆe
´
u
nh`ın t`u
.
d
iˆe
’
m(2a, 0, 0). (DS. −2πa(a + h))
56.


C
(y−z)dx+(z−x)dy+(x−y)dz, C l`a du
.
`o
.
ng tr`on x
2
+y
2
+z
2
= a
2
,
y = xtgα,0<α<
π
2
c´o hu
.
´o
.
ng ngu
.
o
.
.
cchiˆe
`
ukimd
ˆo

`
ng hˆo
`
nh`ın t`u
.
d
iˆe
’
m(2a, 0, 0). (DS. 2
√
2πa
2
sin

π
4
−α))
57.

C
(y −z)dx+(z−x)dy+(x−y)dz, C l`a elip x
2
+y
2
=1,x+z =1
c´o hu
.
´o
.
ng ngu

.
o
.
.
cchiˆe
`
u kim d
ˆo
`
ng hˆo
`
nˆe
´
unh`ınt`u
.
phˆa
`
ndu
.
o
.
ng tru
.
c Oz.
(D
S. −4π)
58.

C
(y

2
−z
2
)dx +(z
2
−x
2
)dy +(x
2
−y
2
)dz, C l`a biˆen cu
’
a thiˆe
´
tdiˆe
.
n
cu
’
alˆa
.
pphu
.
o
.
ng 0  x  a,0 y  a,0 z  a v´o
.
im˘a
.

t ph˘a
’
ng
x + y + z =
3a
2
c´o hu
.
´o
.
ng ngu
.
o
.
.
cchiˆe
`
ukimd
ˆo
`
ng hˆo
`
nˆe
´
u nh`ın t`u
.
d
iˆe
’
m

(2a, 0, 0). (D
S. −
9
2
a
3
)
59.

C
e
x
dx + z(x
2
+ y
2
)
3/2
dy + yz
3
dz, C l`a giao tuyˆe
´
ncu
’
am˘a
.
t z =

x
2

+ y
2
v´o
.
i c´ac m˘a
.
t ph˘a
’
ng x =0,x =2,y =0,y =1.
(D
S. −14)
60.

C
8y

(1 − x
2
− z
2
)
3
dx+xy
3
dy +sin zdz, C l`a biˆen cu
’
amˆo
.
t phˆa
`

n
tu
.
elipxoid 4x
2
+ y
2
+4z
2
=4n˘a
`
m trong g´oc phˆa
`
n t´am th´u
.
I.
176 Chu
.
o
.
ng 12. T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n
(DS.
32
5
)
Chu

.
o
.
ng 13
L´y thuyˆe
´
tchuˆo
˜
i
13.1 Chuˆo
˜
isˆo
´
du
.
o
.
ng 178
13.1.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n 178
13.1.2 Chuˆo
˜
isˆo
´

du
.
o
.
ng 179
13.2 Chuˆo
˜
ihˆo
.
itu
.
tuyˆe
.
td
ˆo
´
i v`a hˆo
.
itu
.
khˆong
tuyˆe
.
td
ˆo
´
i 191
13.2.1 C´ac d
i
.

nh ngh˜ıa co
.
ba
’
n 191
13.2.2 Chuˆo
˜
id
an dˆa
´
u v`a dˆa
´
uhiˆe
.
u Leibnitz . . . . 192
13.3 Chuˆo
˜
il˜uy th`u
.
a 199
13.3.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n 199
13.3.2 D
-

iˆe
`
ukiˆe
.
n khai triˆe
’
n v`a phu
.
o
.
ng ph´ap khai
triˆe
’
n 201
13.4 Chuˆo
˜
iFourier 211
13.4.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n 211
13.4.2 Dˆa
´
uhiˆe
.
ud

u
’
vˆe
`
su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
i Fourier 212
178 Chu
.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
13.1 Chuˆo
˜
isˆo
´
du

.
o
.
ng
13.1.1 C´ac di
.
nh ngh˜ıa co
.
ba
’
n
Gia
’
su
.
’
cho d˜ay sˆo
´
(a
n
). Biˆe
’
uth´u
.
cda
.
ng
a
1
+ a

2
+ ···+ a
n
+ ···=
∞

n=1
a
n
=

n1
a
n
(13.1)
d
u
.
o
.
.
cgo
.
il`achuˆo
˜
isˆo
´
(hay d
o
.

n gia
’
n l`a chuˆo
˜
i). C´ac sˆo
´
a
1
, ,a
n
,
d
u
.
o
.
.
cgo
.
il`ac´ac sˆo
´
ha
.
ng cu
’
achuˆo
˜
i, sˆo
´
ha

.
ng a
n
go
.
il`asˆo
´
ha
.
ng tˆo
’
ng qu´at
cu
’
a chuˆo
˜
i. Tˆo
’
ng n sˆo
´
ha
.
ng d
ˆa
`
u tiˆen cu
’
a chuˆo
˜
idu

.
o
.
.
cgo
.
il`atˆo
’
ng riˆeng
th´u
.
n cu
’
a chuˆo
˜
i v`a k´y hiˆe
.
ul`as
n
,t´u
.
cl`a
s
n
= a
1
+ a
2
+ ···+ a
n

.
V`ısˆo
´
sˆo
´
ha
.
ng cu
’
achuˆo
˜
i l`a vˆo ha
.
n nˆen c´ac tˆo
’
ng riˆeng cu
’
achuˆo
˜
ilˆa
.
p
th`anh d˜ay vˆo ha
.
n c´ac tˆo
’
ng riˆeng s
1
,s
2

, ,s
n
,
D
-
i
.
nh ngh˜ıa 13.1.1. Chuˆo
˜
i (13.1) d
u
.
o
.
.
cgo
.
il`achuˆo
˜
ihˆo
.
itu
.
nˆe
´
u d˜ay
c´ac tˆo
’
ng riˆeng (s
n

)cu
’
an´oc´o gi´o
.
iha
.
nh˜u
.
uha
.
n v`a gi´o
.
iha
.
nd
´odu
.
o
.
.
c
go
.
il`atˆo
’
ng cu
’
achuˆo
˜
ihˆo

.
itu
.
.Nˆe
´
ud˜ay(s
n
) khˆong c´o gi´o
.
iha
.
nh˜u
.
uha
.
n
th`ı chuˆo
˜
i (13.1) phˆan k`y.
D
-
i
.
nh l ´y 13.1.1. D
iˆe
`
ukiˆe
.
ncˆa
`

ndˆe
’
chuˆo
˜
i (13.1) hˆo
.
itu
.
l`a sˆo
´
ha
.
ng tˆo
’
ng
qu´at cu
’
an´odˆa
`
nd
ˆe
´
n 0 khi n →∞,t´u
.
cl`a lim
n→∞
a
n
=0.
D

i
.
nh l´y 13.1.1 chı
’
l`a diˆe
`
ukiˆe
.
ncˆa
`
n ch´u
.
khˆong l`a d
iˆe
`
ukiˆe
.
ndu
’
.
Nhu
.
ng t`u
.
d
´o c´o thˆe
’
r´ut ra diˆe
`
ukiˆe

.
ndu
’
dˆe
’
chuˆo
˜
i phˆan k`y: Nˆe
´
u
lim
n→∞
a
n
=0th`ı chuˆo
˜
i

n1
a
n
phˆan k`y.
Chuˆo
˜
i

nm+1
a
n
thu du

.
o
.
.
ct`u
.
chuˆo
˜
i

n1
a
n
sau khi c˘a
´
tbo
’
m sˆo
´
ha
.
ng
d
ˆa
`
u tiˆen du
.
o
.
.

cgo
.
il`aphˆa
`
ndu
.
th´u
.
m cu
’
achuˆo
˜
i

n1
a
n
.Nˆe
´
u chuˆo
˜
i (13.1)
hˆo
.
itu
.
th`ı mo
.
i phˆa
`

ndu
.
cu
’
an´od
ˆe
`
uhˆo
.
itu
.
,v`amˆo
.
t phˆa
`
ndu
.
n`ao d
´o
hˆo
.
itu
.
th`ı ba
’
n thˆan chuˆo
˜
ic˜ung hˆo
.
itu

.
.Nˆe
´
u phˆa
`
ndu
.
th ´u
.
m cu
’
achuˆo
˜
i
13.1. Chuˆo
˜
isˆo
´
du
.
o
.
ng 179
(13.1) hˆo
.
itu
.
v`a tˆo
’
ng cu

’
an´ob˘a
`
ng R
m
th`ı s = s
m
+ R
m
. Chuˆo
˜
ihˆo
.
itu
.
c´o c´ac t´ınh chˆa
´
t quan tro
.
ng l`a
(i) V´o
.
isˆo
´
m cˆo
´
d
i
.
nh bˆa

´
tk`y chuˆo
˜
i (13.1) v`a chuˆo
˜
i phˆa
`
ndu
.
th ´u
.
m
cu
’
an´od
ˆo
`
ng th`o
.
ihˆo
.
itu
.
ho˘a
.
cd
ˆo
`
ng th`o
.

i phˆan k`y.
(ii) Nˆe
´
u chuˆo
˜
i (13.1) hˆo
.
itu
.
th`ı R
m
→ 0 khi m →∞
(iii) Nˆe
´
u c´ac chuˆo
˜
i

n1
a
n
v`a

n1
b
n
hˆo
.
itu
.

v`a α, β l`a h˘a
`
ng sˆo
´
th`ı

n1
(αa
n
+ βb
n
)=α

n1
a
n
+ β

n1
b
n
.
13.1.2 Chuˆo
˜
isˆo
´
du
.
o
.

ng
Chuˆo
˜
isˆo
´

n1
a
n
du
.
o
.
.
cgo
.
i l`a chuˆo
˜
isˆo
´
du
.
o
.
ng nˆe
´
u a
n
 0 ∀n ∈ N.Nˆe
´

u
a
n
> 0 ∀n th`ı chuˆo
˜
idu
.
o
.
.
cgo
.
il`achuˆo
˜
isˆo
´
du
.
o
.
ng thu
.
.
csu
.
.
.
Tiˆeu chuˆa
’
nhˆo

.
itu
.
. Chuˆo
˜
isˆo
´
du
.
o
.
ng hˆo
.
itu
.
khi v`a chı
’
khi d˜ay tˆo
’
ng
riˆeng cu
’
a n´o bi
.
ch˘a
.
n trˆen.
Nh`o
.
d

iˆe
`
ukiˆe
.
n n`ay, ta c´o thˆe
’
thu du
.
o
.
.
cnh˜u
.
ng dˆa
´
uhiˆe
.
ud
u
’
sau dˆay:
Dˆa
´
uhiˆe
.
u so s´anh I. Gia
’
su
.
’

cho hai chuˆo
˜
isˆo
´
A :

n1
a
n
,a
n
 0 ∀n ∈ N v`a B :

n1
b
n
,b
n
 0 ∀n ∈ N
v`a a
n
 b
n
∀n ∈ N. Khi d´o:
(i) Nˆe
´
u chuˆo
˜
isˆo
´

B hˆo
.
itu
.
th`ı chuˆo
˜
isˆo
´
A hˆo
.
itu
.
,
(ii) Nˆe
´
u chuˆo
˜
isˆo
´
A phˆan k`y th`ı chuˆo
˜
isˆo
´
B phˆan k`y.
Dˆa
´
uhiˆe
.
u so s´anh II. Gia
’

su
.
’
c´ac chuˆo
˜
isˆo
´
A v`a B l`a nh˜u
.
ng chuˆo
˜
i
sˆo
´
du
.
o
.
ng thu
.
.
csu
.
.
v`a ∃ lim
n→∞
a
n
b
n

= λ (r˜o r`ang l`a 0  λ  +∞). Khi
d
´o :
(i) Nˆe
´
u λ<∞ th`ı t`u
.
su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
isˆo
´
B k´eo theo su
.
.
hˆo
.
itu
.
cu
’
a chuˆo

˜
isˆo
´
A
(ii) Nˆe
´
u λ>0th`ıt`u
.
su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
isˆo
´
A k´eo theo su
.
.
hˆo
.
itu
.
cu
’

a chuˆo
˜
isˆo
´
B
180 Chu
.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
(iii) Nˆe
´
u0<λ<+∞ th`ı hai chuˆo
˜
i A v`a B dˆo
`
ng th`o
.
ihˆo
.
itu
.
ho˘a
.
c
d

ˆo
`
ng th`o
.
i phˆan k`y.
Trong thu
.
.
c h`anh dˆa
´
uhiˆe
.
u so s´anh thu
.
`o
.
ng d
u
.
o
.
.
csu
.
’
du
.
ng du
.
´o

.
ida
.
ng
“ thu
.
.
c h`anh” sau d
ˆay:
Dˆa
´
uhiˆe
.
u thu
.
.
c h`anh. Nˆe
´
ud
ˆo
´
iv´o
.
i d˜ay sˆo
´
du
.
o
.
ng (a

n
)tˆo
`
nta
.
i c´ac sˆo
´
p v`a C>0 sao cho a
n
∼
C
n
p
, n →∞th`ı chuˆo
˜
i

n1
a
n
hˆo
.
itu
.
nˆe
´
u p>1
v`a phˆan k`y nˆe
´
u p  1.

C´ac chuˆo
˜
ithu
.
`o
.
ng d
u
.
o
.
.
cd`ung d
ˆe
’
so s´anh l`a
1) Chuˆo
˜
icˆa
´
psˆo
´
nhˆan

n0
aq
n
, a =0hˆo
.
itu

.
khi 0  q<1 v`a phˆan
k`y khi q  1.
2) Chuˆo
˜
i Dirichlet:

n1
1
n
α
hˆo
.
itu
.
khi α>1 v`a phˆan k`y khi α  1.
Chuˆo
˜
i phˆan k`y

n1
1
n
go
.
il`achuˆo
˜
id
iˆe
`

u h`oa.
T`u
.
dˆa
´
uhiˆe
.
u so s´anh I v`a chuˆo
˜
i so s´anh 1) ta r´ut ra:
Dˆa
´
uhiˆe
.
u D’Alembert. Nˆe
´
u chuˆo
˜
i a
1
+ a
2
+ ···+ a
n
+ , a
n
> 0
∀n c´o
lim
n→∞

a
n+1
a
n
= D
th`ı chuˆo
˜
ihˆo
.
itu
.
khi D < 1 v`a phˆan k`y khi D > 1.
Dˆa
´
uhiˆe
.
u Cauchy. Nˆe
´
u chuˆo
˜
i a
1
+ a
2
+ ···+ a
n
+ , a
n
 0 ∀n
c´o

lim
n→∞
n
√
a
n
= C
th`ı chuˆo
˜
ihˆo
.
itu
.
khi C < 1 v`a phˆan k`y khi C > 1.
Trong tru
.
`o
.
ng ho
.
.
p khi D = C = 1 th`ı ca
’
hai dˆa
´
uhiˆe
.
u n`ay d
ˆe
`

u
khˆong cho cˆau tra
’
l`o
.
i kh˘a
’
ng d
i
.
nh v`ıtˆo
`
nta
.
ichuˆo
˜
ihˆo
.
itu
.
lˆa
˜
nchuˆo
˜
i
phˆan k`y v´o
.
i D ho˘a
.
c C b˘a

`
ng 1.
Dˆa
´
uhiˆe
.
ut´ıch phˆan. Nˆe
´
u h`am f(x) x´ac d
i
.
nh ∀x  1 khˆong ˆam
v`a gia
’
mth`ıchuˆo
˜
i

n1
f(n)hˆo
.
itu
.
khi v`a chı
’
khi t´ıch phˆan suy rˆo
.
ng
13.1. Chuˆo
˜

isˆo
´
du
.
o
.
ng 181
∞

0
f(x)dx hˆo
.
itu
.
.
T`u
.
dˆa
´
uhiˆe
.
ut´ıch phˆan suy ra chuˆo
˜
i

n1
1
n
α
hˆo

.
itu
.
khi α>1v`a
phˆan k`y khi 0 <α 1. Nˆe
´
u α  0 th`ı do a
n
=
1
n
α
→ 0 khi α  0v`a
n →∞nˆen chuˆo
˜
id
˜achoc˜ung phˆan k`y.
C
´
AC V
´
IDU
.
V´ı d u
.
1. Kha
’
o s´at su
.
.

hˆo
.
itu
.
cu
’
a c´ac chuˆo
˜
i
1)

n1
1

n(n +1)
;2)

n7
1
n
ln n
·
Gia
’
i. 1) Su
.
’
du
.
ng bˆa

´
td
˘a
’
ng th´u
.
chiˆe
’
n nhiˆen
1

n(n +1)
>
1
n +1
·
V`ıchuˆo
˜
i

n1
1
n +1
l`a phˆa
`
ndu
.
sau sˆo
´
ha

.
ng th´u
.
nhˆa
´
tcu
’
achuˆo
˜
id
iˆe
`
u
h`oa nˆen n´o phˆan k`y.
Do d
´o theo dˆa
´
uhiˆe
.
u so s´anh I chuˆo
˜
id˜a cho phˆan k`y.
2) V`ı ln n>2 ∀n>7nˆen
1
n
ln n
<
1
n
2

∀n>7.
Do chuˆo
˜
i Dirichlet

n7
1
n
2
hˆo
.
itu
.
nˆen suy ra r˘a
`
ng chuˆo
˜
id˜a cho hˆo
.
i
tu
.
. 
V´ı d u
.
2. Kha
’
o s´at su
.
.

hˆo
.
itu
.
cu
’
a c´ac chuˆo
˜
i:
1)

n1
(n − 1)
n
n
n+1
, 2)

n1
n
2
e
−
√
n
.
Gia
’
i. 1) Ta viˆe
´

tsˆo
´
ha
.
ng tˆo
’
ng qu´at cu
’
a c´ac chuˆo
˜
idu
.
´o
.
ida
.
ng:
(n − 1)
n
n
n+1
=
1
n

1 −
1
n

n

.
182 Chu
.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
Ta biˆe
´
tr˘a
`
ng lim
n→∞

1 −
1
n

n
=
1
e
nˆen a
n
∼
n→∞
1

ne
.
Nhu
.
ng chuˆo
˜
i

n→∞
1
ne
phˆan k`y, do d
´o chuˆo
˜
id˜a cho phˆan k`y.
2) R˜o r`ang l`a dˆa
´
uhiˆe
.
u D’Alembert v`a Cauchy khˆong gia
’
i quyˆe
´
t
d
u
.
o
.
.

cvˆa
´
nd
ˆe
`
vˆe
`
su
.
.
hˆo
.
itu
.
. Ta nhˆa
.
nx´et r˘a
`
ng e
−
√
n
=0(n
−
α
2
) khi n →∞
(α>0). T`u
.
d

´o

n1
a
n
=

n1
1
n
a
0
2
−2
hˆo
.
itu
.
nˆe
´
u a
0
> 6. Do vˆa
.
y theo dˆa
´
uhiˆe
.
u so s´anh I chuˆo
˜

i

n1
n
2
e
−
√
n
hˆo
.
itu
.
. 
V´ı du
.
3. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
i

1)

n1
2
n
+ n
2
3
n
+ n
, 2)

n1
(n!)
2
(2n)!
·
Gia
’
i. 1) Ta c´o:
a
n+1
a
n
=
2
n+1
+(n +1)
2
3

n+1
+(n +1)
×
3
n
+ n
2
n
+ n
2
=

2+
(n +1)
2
2
n

3+
n +1
3
n
×
1+
n
3
n
1+
n
2

2
n
,
n
√
a
n
=
2
3
n






1+
n
2
2
n
1+
n
3
n
·
T`u
.
d

´o suy ra lim
n→∞
a
n+1
a
n
=
2
3
v`a lim
n→∞
n
√
a
n
=
2
3
. V`a ca
’
hai dˆa
´
uhiˆe
.
u
Cauchy, D’Alembert d
ˆe
`
uchokˆe
´

t luˆa
.
n chuˆo
˜
ihˆo
.
itu
.
.
2)
´
Ap du
.
ng dˆa
´
uhiˆe
.
u D’Alembert ta c´o:
D = lim
n→∞
a
n+1
a
n
= lim
n→∞
(n +1)
2
(2n + 2)(2n +1)
=

1
4
< 1.
Do d
´o chuˆo
˜
id˜a c h o h ˆo
.
itu
.
.
13.1. Chuˆo
˜
isˆo
´
du
.
o
.
ng 183
Nhˆa
.
nx´et. Nˆe
´
u ´ap du
.
ng bˆa
´
td˘a
’

ng th ´u
.
c

n
e

n
<n! <e

n
2

n
th`ı
(n!)
2
n

(2n)!

1
n
<
e
2
n

n
2


2

2n
e

2
=
e
2+
2
n
4
2
,
do d
´o lim
n→∞
n
√
a
n
<

e
4

2
< 1 v`a khi d´odˆa
´

uhiˆe
.
u Cauchy c˜ung cho ta
kˆe
´
t luˆa
.
n.
V´ı d u
.
4. Kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
i
1)

n1
2n
n
2

+1
, 2)

n2
1
n ln
p
n
,p>0.
Gia
’
i. 1) Ta c´o a
n
=
2n
n
2
+1
= f(n). Trong biˆe
’
uth´u
.
ccu
’
asˆo
´
ha
.
ng
tˆo

’
ng qu´at cu
’
a a
n
=
2n
n
2
+1
ta thay n bo
.
’
ibiˆe
´
n liˆen tu
.
c x v`a ch´u
.
ng to
’
r˘a
`
ng h`am f(x)thud
u
.
o
.
.
cliˆen tu

.
cd
o
.
nd
iˆe
.
u gia
’
m trˆen nu
.
’
a tru
.
cdu
.
o
.
ng.
Ta c´o:
+∞

1
2x
x
2
+1
dx = lim
A→+∞
A


1
2x
x
2
+1
dx = lim
A→+∞
ln(x
2
+1)


A
1
= ln(+∞) −ln 2 = ∞.
Do d
´o chuˆo
˜
i 1) phˆan k`y.
2) Nhu
.
trˆen, ta d
˘a
.
t f(x)=
1
x ln
p
x

, p>0, x  2. H`am f(x) tho
’
a
m˜an mo
.
id
iˆe
`
ukiˆe
.
ncu
’
adˆa
´
uhiˆe
.
u t´ıch phˆan. V`ı t´ıch phˆan
+∞

2
dx
x ln
p
x
hˆo
.
i
tu
.
khi p>1 v`a phˆan k`y khi p  1 nˆen chuˆo

˜
id
˜a cho hˆo
.
itu
.
khi p>1
v`a phˆan k`y khi 0 <p 1- 
184 Chu
.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
V´ı d u
.
5. Ch ´u
.
ng minh r˘a
`
ng chuˆo
˜
i

n1
n +2
(n +1)

√
n
tho
’
a m˜an d
iˆe
`
ukiˆe
.
n
cˆa
`
nhˆo
.
itu
.
nhu
.
ng chuˆo
˜
i phˆan k`y.
Gia
’
i. Ta c´o
a
n
=
n +2
(n +1)
√

n
∼
(n→∞)
1
√
n
⇒ lim
n→∞
a
n
=0.
Tiˆe
´
p theo ∀k =1, 2, ,n ta c´o
a
k
=
k +2
(k +1)
√
k
>
1
√
k

1
√
n
v`a do d

´o
s
n
=
n

k=1
a
k
 n ·
1
√
n
=
√
n → +∞ khi n →∞
v`a do d
´o chuˆo
˜
i phˆan k`y. 
B
`
AI T
ˆ
A
.
P
Trong c´ac b`ai to´an sau d
ˆa y , b ˘a
`

ng c´ach kha
’
o s´at gi´o
.
iha
.
ncu
’
atˆo
’
ng
riˆeng, h˜ay x´ac lˆa
.
p t´ınh hˆo
.
itu
.
(v`a t´ınh tˆo
’
ng S) hay phˆan k`y cu
’
achuˆo
˜
i
1.

n1
1
3
n−1

.(DS. S =
3
2
)
2.

n0
(−1)
n
2
n
.(DS.
2
3
)
3.

n1
(−1)
n−1
.(DS. Phˆan k`y)
4.

n0
ln
2n
2. (DS.
1
1 −ln
2

2
)
5.

n1
1
n(n +5)
.(D
S.
137
300
)
13.1. Chuˆo
˜
isˆo
´
du
.
o
.
ng 185
6.

n1
1
(α + n)(α + n +1)
, α  0. (D
S.
1
α +1

)
7.

n3
1
n
2
−4
.(D
S.
25
48
)
8.

n1
2n +1
n
2
(n +1)
2
.(DS. 1)
9.

n1
(
3
√
n +2− 1
3

√
n +1+
3
√
n). (DS. 1 −
3
√
2)
10.

n1
1
n(n + 3)(n +6)
.(D
S.
73
1080
)
Su
.
’
du
.
ng d
iˆe
`
ukiˆe
.
ncˆa
`

n2)dˆe
’
x´ac di
.
nh xem c´ac chuˆo
˜
i sau dˆay chuˆo
˜
i
n`ao phˆan k`y.
11.

n1
(−1)
n−1
.(DS. Phˆan k`y)
12.

n1
2n − 1
3n +2
.(D
S. Phˆan k`y)
13.

n1
n

0, 001. (DS. Phˆan k`y)
14.


n1
1
√
2n
.(D
S. Dˆa
´
uhiˆe
.
ucˆa
`
n khˆong cho cˆau tra
’
l`o
.
i)
15.

n1
2n
3
n
.(DS. Dˆa
´
uhiˆe
.
ucˆa
`
n khˆong cho cˆau tra

’
l`o
.
i)
16.

n1
1
n
√
0, 3
.(D
S. Phˆan k`y)
17.

n1
1
n
√
n!
.(D
S. Phˆan k`y)
18.

n1
n
2
sin
1
n

2
+ n +1
.(D
S. Phˆan k`y)
186 Chu
.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
19.

n1

1+
1
n

n
2
e
n
.(DS. Phˆan k`y)
20.

n1


2n
2
+1
2n
2
+3

n
2
.(DS. Phˆan k `y)
21.

n1
n
n+
1
n

n +
1
n

n
.(DS. Phˆan k`y)
22.

n1
n +2
(n +1)
√

n
.(D
S. Dˆa
´
uhiˆe
.
ucˆa
`
n khˆong cho cˆau tra
’
l`o
.
i)
23.

n1
(n + 1)arctg
1
n +2
.(D
S. Phˆan k`y)
Trong c´ac b`ai to´an sau d
ˆay, h˜ay d`ung dˆa
´
uhiˆe
.
u so s´anh dˆe
’
kha
’

o
s´at su
.
.
hˆo
.
itu
.
cu
’
a c´ac chuˆo
˜
id
˜a c h o
24

n1
1
√
n
.(D
S. Phˆan k`y)
25.

n1
1
n
n
.(DS. Hˆo
.

itu
.
). Chı
’
dˆa
˜
n. n
n
> 2
n
∀n  3.
26.

n1
1
ln n
.(D
S. Phˆan k`y). Chı
’
dˆa
˜
n. So s´anh v´o
.
i chuˆo
˜
id
iˆe
`
u h`oa.
27.


n1
1
n3
n−1
.(DS. Hˆo
.
itu
.
)
28.

n1
1
3
√
n +1
.(D
S. Phˆan k`y)
29.

n1
1
2
n
+1
.(D
S. Hˆo
.
itu

.
)
30.

n1
n
(n + 2)2
n
.(DS. Hˆo
.
itu
.
)
13.1. Chuˆo
˜
isˆo
´
du
.
o
.
ng 187
31.

n1
1

(n + 2)(n
2
+1)

.(D
S. Hˆo
.
itu
.
)
32.

n1
5n
2
− 3n +10
3n
5
+2n +17
.(D
S. Hˆo
.
itu
.
)
33.

n1
5+3(−1)
n
2
n+3
.(DS. Hˆo
.

itu
.
). Chı
’
dˆa
˜
n. 2  5+3(−1)
n
 8.
34.

n1
ln n
n
.(D
S. Phˆan k`y). Chı
’
dˆa
˜
n. ln n>1 ∀n>2.
35.

n1
ln n
n
2
.(DS. Hˆo
.
itu
.

)
Chı
’
dˆa
˜
n. Su
.
’
du
.
ng hˆe
.
th ´u
.
clnn<n
α
∀α>0v`an du
’
l´o
.
n.
36.

n1
ln n
3
√
n
.(D
S. Phˆan k`y)

37.

n1
n
5
5
√
n
.(DS. Hˆo
.
itu
.
)
38.

n1
1
√
n
sin
1
n
.(D
S. Hˆo
.
itu
.
)
39.


n1
n
4
+4n
2
+1
2
n
.(DS. Hˆo
.
itu
.
)
40.

n1
n
2
(
n
√
a −
n+1
√
a), a>0. (DS. Phˆan k`y ∀a =1)
41.

n1
(
n

√
2 −
n+1
√
2). (DS. Hˆo
.
itu
.
)
42.

n1
1
1+a
n
, a>0. (DS. Hˆo
.
itu
.
khi a>1. Phˆan k`ykhi0<a 1)
43.

n1
sin
πn
n
2
√
n + n +1
.(D

S. Hˆo
.
itu
.
)
Trong c´ac b`ai to´an sau d
ˆay, h˜ay x´ac di
.
nh nh˜u
.
ng gi´a tri
.
cu
’
a tham
sˆo
´
p d
ˆe
’
chuˆo
˜
id˜a cho hˆo
.
itu
.
ho˘a
.
c phˆan k`y:
188 Chu

.
o
.
ng 13. L´y thuyˆe
´
tchuˆo
˜
i
44.

n1

sin
π
n

p
, p>0. (DS. Hˆo
.
itu
.
nˆe
´
u p>1, phˆan k`y nˆe
´
u p  1)
45.

n1
tg

p
π
n +2
, p>0. (D
S. Hˆo
.
itu
.
khi p>1, phˆan k`y khi p  1)
46.

n1
sin
1
n
p
· tg
1
n
q
, p>0, q>0.
(D
S. Hˆo
.
itu
.
khi p + q>1, phˆan k`y khi p + q  1)
47.

n1


1 −cos
1
n
p

, p>0.
(D
S. Hˆo
.
itu
.
khi p>
1
2
, phˆan k`y khi p 
1
2
)
48.

n1
(
√
n +1−
√
n)
p
ln
2n +1

2n +3
.
(D
S. Hˆo
.
itu
.
khi p>0, phˆan k`y khi p  0)
Trong c´ac b`ai to´an sau d
ˆay, h˜ay kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
id
˜a
cho nh`o
.
dˆa
´
uhiˆe
.

ud
u
’
D’Alembert
49.

n1
n
2
n
.(DS. Hˆo
.
itu
.
)
50.

n1
2
n−1
n
n
.(DS. Hˆo
.
itu
.
)
51.

n1

2
n−1
(n − 1)!
.(D
S. Hˆo
.
itu
.
)
52.

n1
n!
2
n
+1
.(D
S. Phˆan k`y)
53.

n1
4
n
n!
n
n
.(DS. Phˆan k`y)
54.

n1

3
n
n2
n
.(DS. Phˆan k `y)
13.1. Chuˆo
˜
isˆo
´
du
.
o
.
ng 189
55.

n1
1 · 3 ···(2n − 1)
3
n
n!
.(D
S. Hˆo
.
itu
.
)
56.

n1

n
2
sin
π
2
n
.(DS. Hˆo
.
itu
.
)
57.

n1
n(n +1)
3
n
.(DS. Hˆo
.
itu
.
)
58.

n1
7
3n
(2n − 5)!
.(D
S. Hˆo

.
itu
.
)
59.

n1
(n + 1)!
2
n
n!
.(D
S. Hˆo
.
itu
.
)
60.

n1
(2n − 1)!!
n!
.(D
S. Phˆan k`y)
61.

n1
n!(2n + 1)!
(3n)!
.(D

S. Hˆo
.
itu
.
)
62.

n1
n
n
sin
π
2
n
n!
.(D
S. Phˆan k `y)
63.

n1
n
n
n!3
n
.(DS. Hˆo
.
itu
.
)
64.


n1
n!a
n
n
n
, a = e, a>0. (DS. Hˆo
.
itu
.
khi a<e, phˆan k`y khi a>e)
Trong c´ac b`ai to´an sau d
ˆay, h˜ay kha
’
o s´at su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜
id
˜a
cho nh`o
.
dˆa

´
uhiˆe
.
ud
u
’
Cauchy
65.

n1

n
2n +1

n
.(DS. Hˆo
.
itu
.
)
66.

n1

arc sin
1
n

n
.(DS. hˆo

.
itu
.
)
67.

n1
1
3
n

n +1
n

n
2
.(DS. Hˆo
.
itu
.
)