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GII HN DY S
http://e-learning.hcmut.edu.vn/
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DY S THC
Dy s l tp hp cc s c nh ch st nh n ln trong tp hp s t nhin N.
VD: 1/ xn = n2, n = 0, 1, 2,
2/ xn= 1/n, n = 1, 2,
3/ {xn} l cp s cng: a, a+d, a+2d,
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Cc cch cho dy s
21, /n nx n x n
1/Dng lit k:
VD: dy 1, 2, 3,; dy 1, 1/2, 1/3,
2/Dng tng minh:
{xn} cho dng biu thc gii tch ca bin n.
3/Dng quy np:
S hng i sau tnh theo cc s hng i trc
VD:
VD: dy 21 11 1, n n nx x x x
dy 11 2 11 1
2
, , n nnx x
x x x
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Dy n iu
{xn} l dy tngxn xn+1, vi mi n ln
{xn} l dy gimxn xn+1, vi mi n ln
Dy tng v dy gim gi chung l dy n iu.
B du = trong nh ngha ta gi l tng(gim) ngt.
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1.Xt hiu s: xn+1 xn(so vi 0)
2.Xt thng s: xn+1/xn(so vi 1)(dng cho dy s dng)
3.Xt o hm ca hm s f(x), vi f(n) = xn
Phng php kho st dy n iu:
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V d
1 11 2
/ :na x n
1 1 / 1 1 :
2nb x
n
gim1 11
1
n
n
x
x n
1
1
1
n nx x
n
0 tng
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2 3/ :
3 4n
nc x
n
Biu thc ging hm s, xt o hm
22 3 1( ) , ( ) 03 4 (3 4)
xf x f x x x
f(x) tng {xn} tng.
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Dy b chn
{xn} l dy b chn trn M : xn M, n N0
{xn} l dy b chn di m : xn m, n N0
{xn} b chn {xn} b chn trn v b chn di
VD: Xet tnh b chan cua cac day
/ 1 nc n / 3nb
2
1/a
n
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1 2 3 4 5 6 7 8, , , , , , , , , ,n nx x x x x x x x x x
DY CON
Cho {xn}, chn ra cc s hng t dy ny
1cch ty theoth t ch s tng dn ta
c 1 dy con ca {xn}.
VD:
{x2n 1}{x2n}
{x2n-1} = {x1, x3, x5, }
{x2n} = {x2, x4, x6, }
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GiI HN DY S
nh ngha n gin: {xn} c gii hn l a
khi n ra tc l xna khi n ln
Dy hit
0 00, : , nN N x a n N
0 00, : nN a x a n N
: limhu hann
n
a x
a
0Nx
0( )
nx n N
a a
nh ngha cht ch:
1x
2x
3x
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V d
lim 11n
n
n 1
1
1 1
n
nx a
n n
1 11 1
1nx n
n
Chng minh
0
1 11 1nn N n n x
Chn N0 1/, vi > 0 ( b)
* Vi = 10-3, tm N0?
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Tnh cht dy hi t
Dy hi t th b chn.an 0 v an a th a 0
an a v a < c th an
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Cc php ton trn dy hi t
lim , lim
lim lim lim
lim lim lim lim 0
lim lim 0 & lim 0n
K :
K :
n nn n
n n n n n n n
n n n n n n n n n
n n n
n n n
x y
x y x y
x y x y y
x x x x
lim tng(hiu, tch, thng, cn,)
=tng(hiu) lim
(hu hn)
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S HOI TU VA DAY CON
VD: dy {xn} = {(1)n} phn k
2
2 1
1 1
1 1V 2 day conn
n
x
x
lim xn
= a Mi dy con caxn
u a
Dy xnphn k 1 day con phan ky
2 day con co lim nhau
2
2 1
nn
n
x ax
x a
aH qu:
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GII HAN KEPCho 3 dy xn, yn, zn
0
lim lim
n n n
n nn n
x y z n N
x z a
lim
nn
y a
n n nx y z
a
Hqu: 0 & lim 0 lim 0n n n n n n
x y n y x
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Dy phn k ra v cng
Giihn = : khng th xt | xn a | !
Dy khng hi t gi l dy phn k:
Khng c giihn
Phn k ra vcng
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0 0lim , : ,> 0n nn
x M N N x M n N
0 0lim , : ,> 0n nn
x M N N x M n N
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V d
lim 2n
n
22 logn M n M
Chng minh
Vi M > 0 (ln) ty ,
Chn N0 > log2M + 1, ta c :
0 2log 2nn N n M M
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Cc php ton trn dy phn k ra
0
lim 0
0( 0),
nn
n
a
a n N
1lim 0n
na limn
n a 1/ Nu th
2/ Nu th limnn
a
()
lim nn
a
,lim nn
b c
lim nn
a
lim ( )
lim ,
n nn
n nn
a b
a b
neu c 0
, lim nn
b
lim ( )n nn
a b
lim n nn
a b
lim nn
a
, lim nn
b
3/
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GII HAN C BAN
3 / lim 1,n
nn
lim 0, 1n
n
na
a
4 / lim 1, 0nn
a a
1 lim
1 1 lim 0
n
n
n
n
a a
a a
0 lim
0 lim 0
n
n
n
n
2/. Ham mu:
1/. Luy tha:
lim 0, 0!
n
n
aa
n
ln5 / lim 0, 0
p
nn
n
ln p nn n a
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2 / lim 1
nn
e n
2 / lim na n
1 21 / lim lim 0n n
b nn
/ lim 2
nn
c 1 / lim 02
n
nd
2
/ lim 03
n
n
nf
V d
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7 DNG V NH
0, 0 , ,
0
0 01 ,0 ,
i vi 4 php ton cng, tr, nhn, chia:
i vi dng m nbna
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2
sin2 / lim 1n
n n
n
1000
3 / lim
n
n n
!1 / lim nn
n
n ! 1 2 10 0nn nn n n n n
2 2
sin
0 01 1
n n n
n n
Vi n 2000: 1000 10 02
n n
n
V d tng hp
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Tng cp s nhn
1 1
2 1 1lim 1 lim
1 1
n qn
n n
qq q q
q q
1 10 0
0 0 0
1
lim lim 1 1
n qn
n n
u q uu u q u q q q
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1 1 14 / lim 1
2 4 2
nn
11 1 2 1lim 2
1 1 2 1 1 2
n
n
2 1 3 9lim ,
3 2 8 32
n
5/ S
0
1 3 3 2 8,
2 2 4 3 21q u S
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25 / lim 1n
n n
2 2
2
1lim
1n
n n
n n
2 2
1 1 1 1lim lim 0. 0
21 1 1n n nn n n
26 / lim 1n
n n n
27 / lim 1 2n
n n
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8 / lim 3 2n n n
n
3
( 0 )
2 3n n 3 2n
3 1
n
3
lim 3nn
x
3 2n n n
nx
28 / lim 3 2n n n
nn
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TIEU CHUAN WEIRSTRASS
Dy tng & bchn trn th hit,
Dy gim & bchndi th hit
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VD: 1/Chng minh tntigiihn sau:
2 2 21 1 1lim 1 ...
2 3 n n
2 2
1 1
1 ... :2nx n
21 1 1 1
21 1
nxn n n n n
1 21
0
( 1)
tangn n nx x x
n
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2/Chng minh tnti v tm giihn dy s:
0 13, 12
nn
xx x
1n nx x
TIEU CHUAN WEIRSTRASS
Dng quy np chng minh xn > 2 (b chn
di)
n iu:
1
21 1 2
2 2n
k
xx
1 02
nx
Gs xk> 2,
12
n nx x
{xn} gim v b chn di nn hi t
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lim nx LGi:
Khi 1lim nx L
Ta li c 1 12n
n
x
x
Qua gii hn khi n, ta c
1 22
LL L
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0
11
n knn
kk
C
n n
2
1 1 12 1 1!
n
kk
n n k
1
2
1 1 22 2 3
1 1 22
n
k
k
B chn:
2
1 1 12
!
n
kk
n n n k
k n
2
1
2 !
n
k k
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PHA DANG VO NH 1
: lim 1 .
na
n
aa en
1 1
1 / lim 1VD :
n
n n e
2 32
2 / lim4
n
n
n
n
2 3
4 4 22
4
2 1lim 1
4Bien oi
nn n
ne
n e
1 .Dang
