BAB
1 BARISAN
R N
R R
n = 1, 2, 3,...
{
}
X n =
{1, 2, 3,...}
X n X :
X
(X n )
(X n : n
N
→R
∈ N)
X n
X
X = (X n : n
∈ N) {X : n ∈ N} n
X =
(( 1)n : n
−
∈ N)
{(−1)
n
: n
∈ N} = {−1, 1}
X = (2, 4, 6, 8,...) X = (2n : n xn+1 = x n + 2 (n
≥ 1)
∈
N
x1 = 2
b
B = (b,b,b,... )
∈ R
b
b
S = (12 , 22 , 32 ,...) =
(n2 : n
∈ N) = (1, 4, 9,...) a ∈ R a =
A = (an : n
1 2
2
3
n
∈ N) = (a, a , a ,...,a ,...) ( , n ∈ N) = 1 2n
( 12 , 41 , 81 , ..., ..., 21n ,...) F = (f n : n
∈ N)
f 1 = 1 f 2 = 1 f n+1 = f n−1 + f n (n
≥ 2)
F = (1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...)
X = (xn )
Y = (yn )
+ Y = (xn + yn : n X +
∈ N) X − − Y = (x − y : n ∈ N) X Y = (x y : n ∈ N) cX = (cx : n ∈ N) n
n
n n
c
X
n
y n = 0, n
X Y
∀ ∈ N = ( : n ∈ N)
Y
xn yn
X X
X
2n
Y Y = ( 11 , 21 , 31 , ..., ..., n1 ,...)
2
X + , ..., ..., 2nn+1 ,...) X + Y = ( 31 , 29 , 19 3
− Y = ( −
1 7 17 2n2 1 , , , ..., .. ., ,...) n 1 2 3
−
X Y = (2, 2, 2, ..., ..., 2,...) 3X = (6, 12, 18, ..., ..., 6n,...) X Y
..., 2n2 ,...) = (2, 8, 18, ..., Z = (0, 2, 0, ..., ..., 1 + ( 1)n,...)
−
X Z
X + Z X
− Z −
X = (xn ) x
(xn )
ε>0
X Z
K (ε)
n
V ε (x) x
≥ k(ε)
xn
X
X = (xn )
x
X
x
K (ε) K
ε ε
K
|x − x|
xn
n
n
≥ K = K (ε)
K (ε)
K x
ε
lim X = 0 atau lim(xn ) = x
→ x
xn xn
→ ∞
x
n
X
x1
x1 = x2
x2 V ε (x2 )
ε > 0
ε V ε (x1 ) ε <
1 2
|x − x | 1
K 1
2
∈ V (x ) x ∈ V (x ) ∩ V (x )
n > K 1
n > K 2
xn
∈ V (x ) 2
ε
n
V ε (x1 )
xn
ε
1
1
ε
2
ε
V ε (x2 )
x1 = x2
x1 = x 2 X = (xn ) x X
∈R x ε V ε (x)
∀n ≥ K (ε)
∈ V (x)
xn
ε
K 2
K (ε)
∀ε > 0, ∃K (ε) ∈ N |x − x| < ε ∀ε > 0, ∃K (ε) ∈ N x − ε < x < x + ε
n
≥ K (ε)
xn
n
∀n ≥ K (ε)
xn
n
( a)
(b)
(b), (c),
d
∈ V (x) ⇔ |x − x| < ε ⇔ −ε < x − x < ε ⇔ x − ε < x
xn
ε
n
n
n
X = (xn )
x
ε V ε (x)
x
V ε (x)
X
ε
x1 , x2 , x3 ,..,xK −1
x
X = (xn ) ε0 > 0
x nk > K
K V ε (x)
xnk
ε K
ε
1 n
1 n
ε> 0 n
≥ k (ε)
< x + ε
K (ε)
1
| − 0| < ε n
ε>0
1 ε
>0
t > 0 K = K (ε) 1 n
≤
1 K
nt
0 <
∈N
1
< ε
K
<ε
n
1
< t
nt
n
≥ K
≥ K
| n1 − 0| = n1 < ε ( n1 ) 1
n2
ε > 0
K
∈ N
n
1
1 ≤ n n 2
n n
1
ε > 0
K
≥ K
1
≥ K
n
≤
1 K
K
<ε
<ε
1
1 ≤ <ε n n 2
( n12 ) 3n+2 n+1
ε > 0
|
3n+2 n+1
− 3| < ε
| 3nn + +12 − 3| = | 3n + n2 +− 31n − 3 | = | n− +11 | = n +1 1 < n1 ε > 0
K n n
≥ K
1
≥ K
n
≤
1 K
<ε
| 3nn + +12 − 3| = | 3n + 2n +− 31n − 3 | = | n− +11 | = n +1 1 < n1 < ε 3n+2 n+1 3n+2 n 1
−
1 K
<ε
| 3nn− +12 − 3| = | 3n + n2 −− 31n + 3 | = | n −5 1 | = n −5 1 untuk n > 1 ε > 0 ε
≥ K > 1
n
5
1
∈ N, K > 1
K
− <
K 1
| 3nn− +12 − 3| = | 3n + n2 −− 31n + 3 | = |n −5 1 | = n −5 1 ≤ K 5− 1 < 5 5ε = ε (1 + ( 1)n ) ε0 = 1
−
K
∈N
n
≥ K
n
|x − 0| = |2 − 0| = 2 > 1 n
(1 + ( 1)n )
−
m
X m
X = (x1 , x2 , x3 ,...,xn ,...) m
m X = (xm+n : n
X
∈ N) = (x
m+1 , xm+2 ,...
)
X = (2, 4, 6, 8, 10, ..., 2n, ...)
2n + 6
X 3
X = (xn : n m
∈ N
X
→
|x − x| < ε n
X m = (xm+n : n
m
(
∈ N)
X m ε > 0
∈ N) X
X = (xn : n = 1, 2, . . .)
(xn ) = x
K n = K, K + 1, K + 2, . . .
X m = xm+n : n = 1, 2, 3, . . .
{
|x
K
←
}
n
m+n
− x| < ε
m+n
≥ K
n = K, K + 1, K + 2, . . .
X m = x
X m = x
X m
ε > 0
|x − x| < ε K = K − m
K m
m+n
n = K m , K m + 1, K m + 2, . . .
|x
m
− x| = |x − x| < ε
m+n m
−
n = K, K + 1 , K + 2, . . .
n
X = x
A = (an )
X = (xn ) C > 0
∈R |x − x| ≤ |a |, ∀n ∈ N x
n
xn
ε A C
≥ K (
)
ε A C
≥ K ( n ≥ m
)
n
n
n
n
0 < na < 1 + na 1 0 ( a1 )( n1 ), n 1+na C = a1 > 0 m = 1 ( 1+1na )
|
0
1 2n
m = 1
− | ≤
0 < n < 2n , n 1 , n N 0 n
| − | ≤ ∀ ∈
∀ ∈ N
0 <
( 21n ) = 0 0<
1 2n
<
1 n
( n1 ) = 0 ( 21n ) = 0
b =
1 1+na 1
C = 1
(bn ) = 0 1 1+a
<
(n) =
∀ ∈ N
0 < b < 1 0 < b < 1
)=ε
( 1+1na )
a > 0 1
ε C ε C
|a | = |a − 0| < |x − x| ≤ C |a | < C (
a > 0
na
N
(an ) = 0 n
n
∈
(an )
≥ m
ε > 0 K A ( εC )
m
n
n
(xn ) = x
x =
≥ K
a =
1 b
− 1
(1 + a )n
a > 0
0 < bn =
1 (1 + a)n
≥ 1 + na
≤ 1 +1 na < na1 ( bn ) = 0
xn+1 = 3xn + 1
x1
z 1 = 3 z 2 = 5 Z n+2 = z n + 2 z n+1
( n21+1 ) = 0 ( 32nn+1 )= +5
3 2
( √ n1+7 ) = 0
√
n ( n+1 )=0
(xn ) = 0 b
( xn ) = 0
| |
( nb ) = 0
∈R xn
√ ( n) = 0
≥ 0, ∀n ∈
N
(xn ) = 0
X = (xn ) M > 0
|x | ≤ M, ∀n ∈ N n
X = (xn )
{x : n ∈ N} n
R
( n1 : n (( 1)n : n
−
∈ N)
∈ N)
M = 1
M = 1
(xn ) K
xm
(2n : n
|x | > K m
K
2m > K
∈ N)
m
m
K
m =
2
K
2
(xn )
ε = 1
(xn ) K (1) N
(xn ) = x
∈ |x − x| < 1 ||x |−|x|| ≤ |x − x| < 1 |x | < |x| + 1 n
≥ K (1)
n
n
n
n
≥ K (1) M = sup{|x |, |x | . . . , |x − |, |x| +1} n
1
|x | ≤ M, ∀n ∈ N
2
k 1
n
(n) X = (n)
M
n = n < M, n x
∈R
| |
∀ ∈ N ∃n ∈ N x < n ((−1) ) X = ((−1) ) x
∈
(n)
x
n
n
a =
(X )
n
|(−1) − a| < 1
ε = 1
∈ N
K 1
n
≥ K
1
R, M
> 0
n
≥ K
n
1
n
|1 − a | < 1
≥
| − 1 − a|
K 1 n
< 1
−2 < a n ≥ K
< 0
1
0 < a < 2
a
X
X = x n )
Y = (yn ) x
− Y X · Y
X
X + Y
∈ R
y
c
cX
x + y x
cx
X = (xn )
x
−y
xy
Z = (z n ) X Z
z = 0
z x z
(X + Y ) = x + y (xn + yn ) = x + y
|(x + y ) − (x + y)| < ε |(x +y )−(x+y)| = |(x −x)−(y +y)| ≤ |x −x|+|y −y| X = (x ) ∀ε > 0 x K |x − x| < n ≥ K n n
n
n
n
n
n
n
n
1
1
Y = (yn )
ε
n
2
y
K = sup K 1 , K 2
{
}
≥ K
n
|(x + y ) − (x + y)| = |(x − x) − (y + y)| ≤ |x − x| + |y − y| n
n
n
n
n
<
ε
2
+
n
ε
2
= ε
ε > 0 X + Y = (xn + yn )
(X + Y ) = x + y
x + y
(XY ) = xy (xn )
M 1 > 0
|x | ≤
M 1
n
{
| |}
max M 1 , y ε>0
n
(xn ) = x n
1
|x − x| < n ≥ K
K 2
≥ K |y − y| < n ≥ K = max {K , K } 1
M =
N
(yn ) = y
K 1
n
∈
n
ε
ε
2M
2
2M
2
|x y − xy| = |(x y − x y) + (xn − xy)| = |x (y − y ) + y (x − x)| ≤ |x ||y − y| + |y||x − x| ≤ M |y − y| + M |x − x| n n
n n
n
n
y
n
n
n
n
n
n
< M (
n
ε ε ) + M ( ) = ε 2M 2M
(cX ) = cx
|cx − cx| = |c||x − x| n
n
| xz − xz | = | x z z −z xz | 1 = |z ||z | |x z − xz | 1 = |z ||z | |x z − x z + x z − x n| 1 = |z ||z | |x (z − z ) + z (x − x)| ≤ |z |x||z | | |z − z | + |z 1| |x − x| n
n
n
n
n
n
n
n
n n
n
n n
z
n
n
n
n
n
n
n
n
n
n
xn
(xn )
|z ||z| n
M >0 n K 1
∈ N
∈N
n
1 2
(z n ) = z 1 2
|z − z | < | n
|x | ≤ M ε = |z | n ≥ K 1
||z | − |z || ≤ |z − z | n
1 2
|z − z | < |z |
n
n
||z | − |z || ≤ |z − z | ⇔ 21 |z | < |z | < 23 |z | ⇒ |z | > 21 |z | n
n
n
n
n
1 2 |z | < |z |
≥ K
1
n
n
≥ K
1
| xz − xz | ≤ |z |x||z | | |z − z | + |z 1| |x − x| < 2|z M | ||z − z | + |z 1| |x − x| n
n
n
n
n
n
2
n
ε > 0
n
(z n ) = z
∈ N |x − x| < | | ε n ≥ K n ≥ K |z − z | < | | ε K = max {K , K , K } | xz − xz | ≤ |z |x||z | | |z − z | + |z 1| |x − x| 2M 1 < z − z | + || |z | |z | |x − x|
(xn ) = x
K 2 , K 3
2
z2 4M
n
1
2
2
3
3
n
n
n
n
n
n
n
2
=
n
z
n
ε
2
+
ε
2
n
= ε
≥ K (an ) (bn )
(z n )
lim((an )(bn ) . . . (z n )) = lim(an )lim(bn ) . . . lim(z n ) k xn )
lim(akn ) = (lim(an ))k ( 2nn+1 ) = 2 ( 2nn+1 ) = 2
2n + 1
=
n
n
1
+
= 2+
n
Y = ( n1 )
X = (2)
(X + Y ) =
2n
X +
1 n
( 2nn+1 ) = X + Y
Y = 2 + 0 = 2 +1 ( 2nn+5 ) = 2
2+ 2n + 1 = n + 5 1+ X = (2 + n1 ) +1 ( 2nn+5 ) =
Z = 1
2 1
1 n
5 n
Z = (1 + n5 )
X = 2
=2
X = (xn )
≥ 0, ∀n ∈ N
x =
xn
(xn )
≥ 0
x<0 ε =
−x
X
x
K
|x − x| < −x ⇔ x < x − x < −x ⇒ x n
n
n
< 0
≥ K
n
≥ 0, ∀n ∈ N x ≥ 0
xn
X = (xn )
Y = (yn )
≤ y , ∀n ∈ N = y − x , ∀n ∈ N Z 0 ≤ (x ) ≤ (y ) xn
z n z n
≥ 0 , ∀n ∈ N
n
(xn )
n
n
n
(yn )
−
≤ (y ) Z = Y − X (xn )
n
X = (xn )
≤ x ≤ b, ∀n ∈ N
a
n
a
≤
(xn )
≤ b
n
X = (xn ) Y = (yn )
Z = (z n )
≤ y ≤
xn
(xn )
∀ ∈N
z n , n
(yn )
(z n )
Y = (yn )
n
(xn )
(z n ) (xn )
(z n )
≥ K
n
w
ε > 0
∈ N
K
|x − w| < ε |z − w| < ε x ≤ y ≤ z , ∀n ∈ N x − w ≤ y − w ≤ z − w, ∀n ∈ N −ε < y − w < ε n ≥ K n
n
n
n
n
n
n
n
n
ε > 0
(yn ) = w
( sinn n ) = 0
−1 ≤ sin n ≤
n
1 1
− ≤ x = − n
n
sin n n
1 n
≤
1
∈ N
n
n
z n = ( n1 ) = 0
yn = sinn n
( sinn n ) = 0 X = (xn )
( xn ) (xn )
| |
|x| =
x x =
|x|
( xn )
| |
||x | − |x|| ≤ |x − x|, ∀n ∈ |x| n
( xn )
| |
N
X = (xn )
x =
( xn )
x > 0
(xn )
0
x = 0
ε>0
≤ √ x < ε n
n
≥ 0
x=0
≥ K n ≥ K
n
x
√ √ ( x )= x
√
≥ 0
xn
n
→ 0 0 ≤ x = x − 0 ≤ ε √ x → 0 ε > 0 xn
n
2
n
n
∈ N
K
√ x
x > 0
n
> 0
√ x − √ x = (√ x −√ √ x)(√ √ x + √ x) = √ x − x√ x + x x + x √ x + √ x ≥ √ x > 0 √ √ ≤ √ n
n
n
n
n
n
1
n
1
xn + x
x
|√ x − √ x| ≤ ( √ 1x )|x − x| n
xn
n
− x → √ 0 √ ( x )= x
→ x
xn
n
(xn ) ( xxn+1 ) n
L =
L < 1
(xn )
(xn ) = 0 (xn )
L
L
r ε = r
− L > 0 |
xn+1 xn
∈ N
K
− L| < ε
≥ K
n xn+1 xn
≥ K
n n
≥ 0
≥ K
< L + ε = L + (r
− L) = r
0 < xn+1 < xn r < xn−1 r2 < .. . < xK rn−K +1 C =
0 < r < 1 (xn ) = 0
xn =
( xxn+1 )= n
1 2
L<1
xK rK n
0 < xn+1 < Cr n+1, n
∀ ≥ K
(r ) = 0
n
2n
xn+1 xn
( 2nn ) = 0 2n = 2nn+1 = 21 (1+ 1n ) +1 n
·
( 2nn ) = 0
X X
Y X + Y
Y
(2n )
((2 + n1 )2 )
√ n−1 ( √ n+1 ) yn =
√ n + 1 − √ n 0 < a < 1
n
√
(yn )
∈ N
( nyn )
b > 1
(an ) n
( 2bn )
X = (xn )
|x − x| n
C
(an ) X
X
(xn )
x1
≤ x ≤ . . . ≤ x ≤ . . .
xn
x1
≥ x ≥ . . . ≥ x ≥ . . .
xn
2
n
2
n
≤ x
n+1
n
∈N
≥ x
n+1
n
∈N
(1, 2, 3, 4, . . . , n , . . . ) (1, 2, 2, 3, 3, 3, 4, 4, 4, 4, . . .) (1, 21 , 31 , . . . , n1 , . . .) (a, a2 , a3 , . . . , an , . . .) a < 0 a > 0 ( 1, +1, 1, . . . , ( 1) n , . . .) (2, 2, . . . , 2, . . .) (7, 6, 2, 1, 2, 3, 4, . . .) ( 2, 0, 1, 13, 12, 13, . . .)
−
−
−
−
X = (xn )
X = (xn )
X = (xn )
X = (xn )
X = (xn )
X
{x {x
n
n
: n
∈ N} : n ∈ N}
X
X
X X
X
X X
X
M xn
≤ M, ∀n ∈ N x∗ = sup xn : n
{
∈ R, m > 0 {x : n ∈ N} n
∈ N}
(xn ) = x∗ x∗
ε > 0
{x
n
:n
∈ N} x∗ − ε < x
−ε
∈ N, K = K (ε)
K
xn
K
x∗
− ε < x < x < x∗ |x − x∗| < ε, ∀n ≥ K K
n
ε
n
x∗
(xn )
X Y =
X
} N} N
{−x
:n
∈ N} {x : n ∈ N} {x : n ∈ N}
n
Y
Y
{−x {−x
X
X
n
−X n
Y
n
:n :n
∈ ∈
n
1 ) X = ( √ n
√ n + 1 > √ n
√ n1+1 < √ 1n
X M = 1
X xn < 1
xn = 1 +
1 2
n
1 3
+
X X
+ . . . + n1 , n
X = x n
X
xn+1 = x n +
∈ N
1 n+1
∀ ∈ N
> xn
n
n
∈N n
xn = 1 +
1 2
+ 31 + . . . +
1 n
x10 = 2, 9290 x100 = 5, 1874 x1000 = 7, 4855 x10000 = 9, 7876 x100000 = 12, 0901
n
2n
x2n
n = 1 x21 = 1+ 12
n = 2 x22 = 1 + 21 + ( 13 + 41 )
( 15 + 61 +
1 7
n = 3 x23 = 1 + 21 + ( 13 + 41 ) +
+ 81 )
1 1 1 1 + ( + ) + . . . + n ) 2 3 4 2 1 1 1 1 1 1 = 1+ +( 1 + 2 ) + . . . + ( n−1 + n−1 + . . . + n ) 2 2 +1 2 2 +1 2 +2 2
x2n = 1 +
sebanyak n partisi
1 1 1 1 1 1 + ( + ) + . . . + ( n + n + . . . + n ) 2 4 4 2 2 2 n 1 1 1 1 = 1 + + + + . . . + = 1 + 2 2 2 2 2 > 1 +
M = 5001
210000 (xn )
x1 = 1 xn+1 =
√ 2x , untuk n ≥ 1 n
x1 = 1 x2
√
√ 2x = 2 2 2 1 ≤ x ≤ x ≤ x ≤ x ≤ 2 x4 =
3
1
2
3
√ = 2x
1
→ ∞
n
xn =
4
1
√ = 2
x3
√ √ = 2x = 2 2 √ 2 2 . . . = 2
2 2
2
≤ x ≤ x ≤ 2 n
n+1
n=
1, 2, 3, 4
n = k
1
≤ x ≤ x ≤ 2 k
k+1
n = k + 1
1 1
≤ x ≤ x ≤ 2 k+1
k+2
≤ x ≤ x ≤ 2 ⇔ 2 ≤ 2x ≤ 2x ≤ 4 √ ⇔ 1 < 2 ≤ x = √ 2x ≤ k+1
k
k+1
k
k+1
1
k
2xk+1 = x k+2
≤ x ≤ x ≤ 2 k+1
≤ 2
n = k + 1
k+2
{x } {x } n
(xn )
n
x1 = 1 x2 = 1, 4142 x3 = 1, 6818 x4 = 1, 8340 x5 = 1, 9152 x6 = 1, 9571 x7 = 1, 9785 x8 = 1, 9892 x9 = 1, 9946
x10 = 1, 9973
x =
√
lim(xn+1 ) = lim( 2xn ) = x =
√
2x
lim(2xn )
x2 = 2x x(x x = 0
− 2) = 0 x = 2
xn > 1
x= 2
(xn ) = x =
√
√
2 2 2 2...
2 2 2 2... x2 = 2x
⇒ x(x − 2) = 0 ⇒ x = 0 atau x = 2
(xn )
(xn ) en = (1 +
1 n n
)
n
E = (en )
∈N
E = (en )
en = (1 +
1 n
1 n(n − 1) 1 n(n − 1)(n − 2) 1 + + · · ·n n 1 n 2! 3! n(n − 1) . . . 2 · 1 1 + . . . + ·
)n = 1 +
n
2
nn
n!
− n1 ) + 3!1 (1 − n1 )(1 − n2 ) n−1 1 1 2 + . . . + (1 − )(1 − ) . . . (1 − ) n! n n n =1+1+
1 (1 2!
3
n + 1
− n +1 1 ) + 3!1 (1 − n +1 1 )(1 − n +2 1 ) n−1 1 1 2 + . . . + (1 − )(1 − ) . . . (1 − ) n! n + 1 n + 1 n + 1 n 1 1 2 + (1 − )(1 − ) . . . (1 − ) (n + 1)! n + 1 n + 1 n + 1
en+1 = 1 + 1 +
1 (1 2!
n + 1
en
en+1
n + 2
en en+1
2
en
en+1
≤ e < e < .. . < e 1
2
n
< en+1 < . . .
E p = 1, 2, . . . , n
(1
−
p n
2 p−1
) < 1 1 p!
n > 1
2 < en < 1 + 1 +
≤
≤ p!
1 2p−1
1 1 1 + 2 + . . . + n−1 2 2 2
E
n
(sn = a( 11−−rr )) 1 1 1 + 2 + . . . + n−1 = 1 2 2 2 2
≤ e
− 2 1−
n
< 1
n 1
< 3
∈ N
n
E e
e
e e en
x1 = 8
n
xn+1 = 21 xn + 2
n
(xn ) x1 > 1
≥ 2
x1
(xn )
x1 = 1
xn+1 = 2
−
1
n
xn
√ − 1
xn+1 = 1+ xn
xn+1 =
√ 2 + x
(xn )
∈N ∈ N
n
∈ N
n
n
(xn ) x1 = a > 0
∈ N
xn+1 = x n + x1n
n
(xn ) A
∈N R
u = sup A
(xn )
∈ A
u =
∈ N
xn
n
(xn )
(xn )
{x
k
(tn )
: k
≥ n}
∈ N
n tn =
{x
k
: k
≥ n}
(sn )
(tn )
sn =
(sn )
(xn )
(sn )
(xn )
(tn )
(xn ) yn =
1
+
n+1
1 n+2
+ . . . +
1 2n
n
(yn ) xn =
1 + 212 + . . . + n12 12
∈ N (xn )
∈ N
n
≥ 2
k
1
k2
≤
1
1 1 − = k−1 − k
k(k 1)
((1 + n1 )n+1 ) ((1 + n1 )2n ) 1
((1 +
n+1
((1
n
−
)n )
1 n
) )
X = (xn ) X ∗
n1 < n2 < .. . < nk < . . .
(xn1 , xn2 , xn3 , . . . , xnk , . . .) X
X ∗ = (xnk : k X = (1, 21 , 31 , . . . , n1 , . . .)
X X ∗ = ( 12 , 41 , . . . , 21n , . . .) X ∗∗ = (1, 31 , 51 , . . . , 2n1−1 , . . .) 1 X ∗∗∗ = ( 14 , 51 , 61 , . . . , n+3 , . . .)
X Y ∗ = (1, 31 , 21 , 41 , 31 , . . .) Y ∗∗ = ( 12 , 21 , 31 , 31 , . . .)
∈ N)
(xn )
x
x ε > 0
(xn )
x
K (ε)
|x − x| < ε
n
n
≥ K (ε)
n1 < n2 < . . . < n k < . . . nk n1
≥ k +1
nk+1 ε
≥1
nk
≥k k ≥ K (ε)
≥k
≥k |x − x| <
nk+1 > nk
≥ k ≥ K (ε)
nk
(xnk )
nk
x
( bn ) = 0
0 < b < 1 xn = bn xn = bn
0 < b < 1 xn+1 = bn+1 < bn = xn
0
0
x =
(xn ) x2n = b 2n = (bn )2 = (xn )2
(xn ) (x2n )
x =
x = lim (x2n ) = [lim (xn )]2 = x2
0=
(xn ) 1 xn
≤ ≤
(x2n )
2
⇔ x − x = 0 ⇔ x = 0 atau x = 1
(xn ) (xn )
x= 1
(c n ) = 1 1
z n = c n cn+1 > cn
cn+1 > cn
c > 1 n+1
c > 1
1
n
⇔ (c ) > (c ) ⇔ c = z > c = z ⇔ c > c ⇔ c > 1 n(n+1)
1
1
n
n+1 n
1
n
1
n(n+1)
n
n+1
n+1
z n > 1 z n+1 > 1
n
z 1 = c > z n >
(z n )
∈ N
z =
(z n )
z = 1
1
(z 2n )
1
1
z 2n = c 2n = (c n ) 2 = (z n ) 2
1
1
z = lim (z 2n = (lim (z n )) 2 = z 2 z 2 = z n
z = 0
z = 1
z = 1 =
∈N
(z n )
(xn ) (xn )
z n > 1
x
x
(( 1)n )
−
X = (xn ) = (( 1)n )
−
X ∗ = (x2n ) = (( 1)2n )
−
(( 1)2n−1 )
−
X ∗
X ∗∗ = (x2n−1 ) =
X ∗∗
=
(( 1)n )
−
(1, 21 , 3, 41 , . . .) Y = (yn ) yn = n
n
yn =
1
n
n
(1, 3, 5, 7, . . .) ( 12 , 41 , 61 , . . .)
Y
(1, 3, 5, 7, . . .)
Y Y
X = (xn )
m xm
xm
≥x
n
n
≥m
X xm1 , xm2 , . . . , xmk , . . .
m1 < m2 < . . . <
mk < . . . xm1
(xmk )
≥ x ≥ . . . ≥ x ≥ . . . m2
mk
X
X xm1 , xm2 , . . . , xmr s1 = m r + 1 xs1 xs2
s2 > s1 s3 > s2
xs2 > xs1 xs3 > xs2
(xsn ) X
X =
(xn )
X ∗ = (xnk )
X
X ∗ = (xnk )
(( 1)n )
−
(x2n ) = ( ( 1)2n )
(( 1)2n−1 )
−
(x2n−1 ) =
−
X ∗
X ∗
X X ∗1 X ∗1
X
X
x X
X
∈ R
x
x M > 0, M
X
|x | ≤ M
n
n
∈N
X
∈
R
x X ∗ = (xtn )
ε0 > 0
X
|x − x| ≥ ε , untuk semua n ∈ N 0
tn
X ∗
X ∗
X
M X ∗
X ∗∗
X ∗∗
X
X ∗∗
x
X ∗∗
X ∗∗
X ∗ ε0
X ∗ X ∗∗
x X ∗∗
x
0 < c < 1
1
(c n ) X = (xn ) Y = (yn )
Z = (z n )
z 1 = x1 , z 2 = y1 , . . . , z2 n−1 = xn , z 2 n = yn , . . .
Z
X
Y
X
Y
X = (xn ) X
(xn ) sup xk : k
{
∈ N s
n s = inf sn : n
≥ n}
{
(xn ) xn
≥0
∈ N}
=
s
n
(xn )
n
(( 1)n xn )
∈N
−
(xn ) ( xn1 ) = 0
(xnk )
k
(xn )
s = sup xn : n
∈ {x
s
n
{
:n
∈ N}
s
∈ N}
(xn )
X = (xn ) ε > 0
K (ε)
|x − x | < ε n
m
m, n
≥ K (ε)
( n1 ) ε > 0 K >
2 ε
K = K (ε) m, n
≥ K
1 n
≤
1 K
<
m, n
ε
1
2
m
<
ε
2
≥ K | n1 − m1 | ≤ n1 + m1 < 2ε + 2ε = ε ε>0
( n1 ) (1 + ( 1)n )
−
ε0 > 0 K
n > K
|x − x | ≥ ε n
n
m
m > K xn = 1+( 1)n
−
0
xn = 2
xn+1 = 0
K
ε0 = 2
n > K
m = n + 1
|x − x | = |x − x | = |2 − 0| = 2 = ε n
m
n+1
n
0
(xn )
m
|x − x | < ε
n n, m
n
≥ K (ε)
m
n
m
|x − x | ≥ ε n
m
X = (xn ) x = n, m
≥ K (
ε > 0 n
ε
2
0
X
X
K ( 2ε )
n
)
≥ K (
ε
2
)
|x − x| < n
ε
2
|x − x | = |(x − x) + (x − x )| ≤ |x − x| + |x − x| < 2ε + 2ε = ε n
m
n
n
m
m
m
ε > 0 xn
| − x | < ε
(xn )
n, m
m
≥ K
X = (xn ) ε = 1
K = K (1)
n
|x − x | < 1 |x | ≥ |x | + 1
≥ K
n
n
n
≥ K
K
K
M = sup x1 , x2 , . . . , xK −1 , xH + 1
{| | | |
|
|| | }
|x | ≤ M
n
n
∈N
X = (xn ) X X ∗ = (xnk )
X
x∗ x∗
X X = (xn ) K 1 = K 1 ( 2ε )
ε > 0
∈N
≥ K
m, n
1
|x − x | < 2ε n
m
X ∗ = (xnk ) K 2 = K 2 ( 2ε )
x∗
∈N
nk
≥ K
|x − x∗| < 2ε nk
K = max K 1 , K 2
{
}
2
|x − x | < n
m
ε
ε
|x − x∗| < nk
2
≥ K
2
n,m,nk
m = K = n k
|x − x | < n
K
ε
|x − x∗| < n ≥ K K
2
ε
≥ K
n
2
|x − x∗| = |x − x + x − x∗| ≤ |x − x | + |x − x∗| < 2ε + 2ε = ε n
n
K
n
K
K
K
x∗
(xn )
(xn )
x1 = 1, x2 = 2 xn = 21 (xn−1 + xn−2 ), untuk n
≥ 3 1 ≤ x ≤ 2 n
n
∈N
x1 = 1.0000 x2 = 2.0000 x3 = 1.5000 x4 = 1.7500 x5 = 1.6250 x6 = 1.6875 x7 = 1.6563 x8 = 1.6719 x9 = 1.6641
x10 = 1.6680
|x − x | = |x − ( 12 (x + x − ))| 1 = |x − x − | 2 1 = |x − − x | 2 1 1 = |x − − x − | = |x − − x − | 2 2 n
n+1
n
n 1
n
n 1
2
=
n 1
n
n
n 1
n 2
2
1
1 | − | = x x 2 − 2 − n 1
2
1
n 1
n 2
n 1
n, n + 1, n + 2, . . . , m
m > n
− 1, m
|x − x | = |(x − x ) + (x − x ) + . . . + (x − − x )| ≤ |x − x | + |x − x | + . . . + |x − − x | n
m
n
n
n+1
n+1
n+1
n+1
n+2
m 1
n+2
m 1
m
m
1 1 . . . + + + 2n−1 2n 2m−2 1 1 1 = n−1 (1 + + . . . + m−n−1 ) 2 2 2 1 1 2 1 = n−1 (2 ( )m−n−1 ) < n−1 = n−2 2 2 2 2 =
1
−
(2
K
|x − x | < ε n
x =
1 (x 2
m, n
m
2
−
log ε)
≥ K
+ x)
(x2n+1 : n x4 = 21 (2 + 23 ) = (1 + 1 (x3 + x 4 ) = 1 + 21 + 213 2 x7 = 1 + 21 + 213 + 215
∈ N) 1 2
n = 1
+ 41 ) x6 = 1 +
x3 = 1 + n = 2
1 2
+
1 23
+
1 24
1 1 1 1 + 3 + 5 + . . . + 2n−1 2 2 2 2
= 1+
1 (1 2
2 = 1 + (1 3
deret geometri n suku
−( 3 4
−
1 n ) ) 4
1n ( )) 4
2 lim(xn ) = lim(x2n+1) = lim(1 + (1 3
−
x5 = n = 3
n x2n+1 = 1 +
1 2
1n 2 5 ( ))) = 1 + = 4 3 3
X = (xn )
0 < C < 1
C
|x − x | < C |x − x | n+2
n+1
n+1
n
n
dn = xn+1
|
− x |
C
n
(dn )
(xn )
(xn )
(xn )
|x − x | 2
1
|x − x | ≤ C |x − x | ≤ C C |x − x − | = C |x − x − | = C C |x − − x − | = C |x − − x − | n+2
n+1
n+1
n
2
n 1
n
2
n 1
n
n 1
n
3
n 2
n 1
n 2
≤ C |x − x | 2
1
|x − x | n
m>n
m
|x − x | = |(x − x ) + ( x − x ) + (x − x ) + . . . + (x − − x )| ≤ |x − x | + |x − x | + |x − x | + . . . + |x − − x | = |x − x | + |x − x | + |x − x | + . . . + |x − x − | ≤ (C − + C + C + . . . + C − ) |x − x | n
m
n
n
n+1
n+1
n+2
n+3
m 1
n+1
n+2
n+2
n+3
m 1
n
n+2
n+1
n+3
n+2
m
n+1
n
m 2
(m n) deret suku geometri
−
1 − C m−n − = C ( )|x2 − x1| n 1
1 1
− C ≤ C − ( 1 − C )|x − x | → n 1
n+2
n+1
n 1
n+1
2
1
0
2
1
m
m
m 1
0
(xn )
X = (xn ) x∗ =
C, 0 < C
C n−1 1 C
n
1 C
|x∗ − x | ≤ |x∗ − x | ≤
X
− |x2 − x1 |
C − |xn − xn−1 |
|x −
m>n xm = xm xn
(i)
| | − |≤
n−1
C
− |x2 − x1 |
m
1 C
n
→∞
(ii) m>n
|x − x | = |x − x | ≤ C − ( 1 −1 C )|x − x | 1 1 =( )C − |x − x | = ( )|x − x − | 1 − C 1 − C n
m
m
n
n 1
n 1
1
2
1
n
n 1
(ii)
→ ∞
m
2
x1 = 1, x2 = 2 xn = 21 (xn−1 + xn−2 ), untuk n
≥ 3
|x − x | = | 12 (x + x ) − x | 1 1 =| x − x | 2 2 1 = |x −x | 2 n+2
n+1
n+1
n
n+1
n
n+1
C = X = (xn )
1 2
n+1
n
1 2
|x − x | = |x − x | n+2
n+1
n+1
n
x1 = 2
xn+1 = 2 + x1n
≥ 1
n
(xn ) (xn ) x1 = 2 xn = 2 +
1 xn−1
xn+1 = 2 +
1
xn > 0
xn
1
>2
xn
<
1 2
n
≥ 1
|x − x | = | x 1 − x1 | x −x =| | xnx ≤ 41 |x − x | = 41 |x − x | n+2
n+1
n+1
n
n+1
n
n+1
n+1
n
n+1
n
(xn ) x = x2
x = 2+
X
− 2x − 1 = 0
1 x
±√ √ 2 (x ) = 1 + 2 x1,2 = 1
xn > 0 , n
∀ ∈N
n
0 < x1 < 1
x1
xn+1 =
0 < x1 < 1 n
1 3 (x + 2), n 7 n
xn =
≥ 1
1 (x3n 1 7
− + 2) <
∈N
|x − x | = | 17 (x + 2) − 17 (x + 2) | 1 1 = |x − x | = |(x + x 7 7 ≤ 73 (x − x ) n+2
3
n+1
C =
3 7
< 1
< 1
3
n+1
n
3
3
n+1
n
n+1
3 7
n
2
n+1
n+1 xn
+ x2n )(xn+1
− x )| n
x = lim (xn ) x3
− 7x + 2 = 0 x3 x
− 7x + 2 = 0
x1
x1 = 0, 5
x2 , x 3 , . . .
x1 = 0.5 x2 = 0.303571429 x3 = 0.289710830 x4 = 0.289188016 x5 = 0.289169244 x6 = 0.289168571
(xn )
≈ 0, 289169 |x − x | < 0, 2 2
n
|x∗ − xn| ≤
1
3n 1 7 1 73 n 1
−
− · 0, 2 3 − 7 2 = − · · 7 4 10 n 1
3n−1 = n−2 7 20
·
|x∗ − x | = |0, 289169 − 0.289168571| ≈ 0, 0000004 6
n = 6
35 243 = (74 20) 48.020
·
≈ 0.0051
|x∗ − x | = |0, 289169 − 0.289168571| ≈ 0, 0000004 ≤ (7 3· 20) = 48243 ≈ 0.0051 .020 6
5
4
( n+1 ) n (1 + 2!1 + . . . +
1 n!
)
(( 1)n )
−
n (n + (−n1) )
(ln n) (xn ) (xn + yn )
xn =
√ n
(xn yn )
(xn )
0 < r < 1 (xn )
|x − x | < r n+1
n
|x − x | = 0 n+1
n
n
n > 2
(xn )
x1 > 0
(yn )
xn+1 = (2 + xn )−1
1 (xn 2 2
− +
yn = 31 yn−1 + 32 yn−2 )
y1 < y2 n > 2
n
∈ N xn =
x1 < x2 xn−1 )
(yn )
≥ 1
n
(xn )
