Suites numériques
Short Description
Description : Suites numériques...
Description
(un )
(vn )
<
un =
un < v n
un =
(un )
∈Z
N
(un )
1 n
√
un = 1 +
n
n2 + n + 1 n
1
k
n2
(a, b)
∈
(un )
(vn )
un
→a
vn
(vn ) (un )
n N, un a vn un + vn a+b
∈
b
1/n
un =
(un + vn )
(un
(vn )
(vn )
sin n n+( 1)n+1 n ( 1)n n+( 1)n
un =
−v
n)
−
n
un = un =
S n =
lim max(u max(un , vn )
→+∞
n
(vn )
S n =
→0
n
k
k =1 n
S n =
n
nn
−
√ −
k =1 n
u2n + un vn + vn2
n! nn
2 + ( 1)n
n
S n =
k =1
1
k =1 n
(vn ) (un )
n 1 n+1
(un ) un =
S n = (un )
→
→b
n
(un )
n
(un )
− un = − − − (un )
n
√
√n2 +1 un = nn− + n2 −1
k =1
un = sin n1 R2
−− −
− √n2 − n + 1 √2 un =
(un )
3n ( 2)n 3n +( 2)n
un =
n2 +k 2
S n =
n n2 +k 2
S n =
√1k 2n
1
k=n+1 n k =1
( 1)n−k k!
k2
√n1 +k 2
k =0
(un )
(vn )
0 un
1 0 vn
1
un vn
→1
lim
lim
→+∞ n→+∞
m
1 m
− 1
n
n
lim
lim
→+∞ m→+∞
1 m
− 1
n
lim
→+∞
n
1 n
− 1
n
√u →
(un )
n
< 1 > 1
un un =1
→0 → +∞
n
z
|z | < 1
∈C
n
lim
→+∞ k=0
n
un+1 un
(un ) < 1 > 1
un un =1
→0 → +∞
(un )
n
S n =
∈N
k =1 p p+1 x x p
p > 1 (S n ) S 2 n = S n
n
a
n
∈R sin
a
2n
∈N
P n =
P n =
1 2n
k=1
S n =
(vn ) v v2n u + 2 vn
−
k
k =1
n
···+u
n
n
n
vn = sup u p
(un )
vn = sup u p
k
wn = inf u p pn
pn
(vn )
lim P n
(wn )
R
n
∞
n k
−1 n
n
n
n
lim S n
u1 +
pn
cos 2a
vn =
(un )
−1
n+p n∈N u = ∈ N\ {0, 1} n 2)u +2 = (n ( n + 2)u 2)u +1 ∀n ∈ N (n + p + 2)u 1 S = − (1 − (n + p + 1)u 1)u +1 ) 1 ( n + p)u (v ) ∀n ∈ N v = (n n
.
p p 1 p p−1 xx
H = ∈N ∀n ∈ N , H 2 − H n
p
→
( 1)k−1
(S n )
k=0
n
sin a
n
un =
1 n+k
k
→
n
n
1 + z2
n
n
p
n
k =1
n
1 k
1 2
lim H n = +
∞
n
∞
n
S n =
uk
k=1
(H n )
n H n
n
p
n
(un )
→ +∞
∈N
n(un+1
n
H n =
k =1
−u )→1 n
1 k
un
→ +∞
3×5×···×(2n−1) un = 1× 2×4×6×···×(2n) un (un ) vn = (n ( n + 1)u 1)u2n
∈ R+2 ∀n ∈ N, u +1 = √u (a, b)
(vn )
lim un
n
(un )
√
2 ab a + b (un ) (vn ) u +v n vn , vn+1 = 2 n 1 un vn un un+1 vn+1 (vn ) n
u0 = a, v0 = b
n
vn
a
θ
]0, π/2[ ∈ ]0,
n
θ
n
M (a, b) M ( M (a, a) M ( M (a, 0) M ( M (λa,λb) λa,λb)
θ
n
un = 2 sin 2 vn = 2 tan 2 (un ) (vn ) n
S n =
k=1
1 k2
S n = S n +
(S n )
(S n )
(un )
n
S n =
∈N
λ
∈ R+
(un ) (un )
1 n
(u2n )
π2 6
(un ) (un )
n
∈ R+
M ( M (a, b)
n
n
∈N
a
(u2n ), (u2n+1 )
(u3n )
( 1)k uk
−
k =0
(S 2n )
cos n
(S 2n+1 )
(S n )
sin n n
an =
k =0
1 k!
n
bn = (an )
k =0
1 k!
+
1 n.n!
= an +
1 n.n!
(bn )
= aq <
< bq
p q
p
∈/ Q
∈ Z, q ∈ N
un
(un ) 0
→
∀n, p ∈ N
0 un+ p
n+ p
np
b
(un ), (vn ), (wn ), (tn ) wn 1 n
1
, n2 ,
ln n ln n n
,
n2
,
√ n, n , n ln n, n ln n,
1 n ln n
2
∼t
un
n
un + wn
∼v
n
+ tn
2
un =
2
2n3 ln n+1 n2 +1
−
un = (n ( n + 3ln n) −(n+1)
1
n!+ n 2n +3n
ln(n2 +1) n+1
x
un = 1
−
un = cos n1
ln(n ln(n + 1)
un = n ln 1 +
1
n2 +1
ln(n) − ln(n
R
un = 1 + sin
1 n n
un =
x
=n
n
+
xn
∈N
(x n )
n
(un )
n
√
n
un un = ln sin n1
0 (un )
∼1
n2 +n+1 un = √ 3 n2 −n+1
√ (u √) = n+1− n−1
1 n+1
un = n−1 un = sin √n1+1
−
un =
un =
un + un+1
+
(un )
(un )
−√n +1 ln n−2n
n3
n
n2 ln n
(un )
un =
∼v
√ n n+1 √ n
(n+1)
E n E n (xn )
x + ln x = n
E n E n (xn )
x + tan x = n
x xn
∈ R+
+ (x n )
∞
n
n
un = 0! + 1! + 2! +
∈N n
S n =
k =1
√1k
√n1+1
un = S n
2
√
n+1 (S n ) 2 n
− √
· · · + n! =
− √n
(S n )
k =0
k!
un ! n
√1n n
(un )
E n E n (xn )
xn ln x = 1 xn
x xn
/2,, π /2[ ∈ ]−π/2
xn
x 1
∈ R+
n xn
∈N
E n : xn + xn−1 + E n
1 2, 1
∈
(un )n0 u0 = 0 u0 = 0
(xn ) yn+1 = x
n
zn
··· + x =1
(x n ) (xn )
(zn )n0
n
n
u0 = 0, 0 , u1 = 1 + 4i 4i
n
un
(yn )
∀n ∈ N, x +1 = n
+yn 2
xn
−y
(un )n0 (un )n0 (un )n0
zn = xn + i.yn
∀n ∈ N, z +1 = 13 (z n
(zn )
n
+ 2¯zn ) z0
θ
u0 = 1 v0 = 2
θ
]0, π [ ∈ ]0,
2
n
n
n
−
4 u +1 − 4u ∀n ∈ N, u +2 = 4u 3 u +1 − u ∀n ∈ N, 2u +2 = 3u ∀n ∈ N, u +2 = u +1 − u n
n
n
n
n
n
n
n
n
(un )
∈R ∀n ∈ N, u +2 − 2cos θu n
n+1
+ un = 0. 0.
∈N
−
(un )
ρ> 0
n
vn+1 = 2u 2 un + 3v 3vn (un vn )
(un )
| |
n
∀n ∈ N, u +2 = (3 − 2i)u +1 − (5 − 5i)u
u0 = 1, 1 , u1 = 0 u0 = 1, 1 , u1 = 1 u0 = 1, 1 , u1 = 2
u0 = u1 = 1
(un ) (vn ) un+1 = 3u 3 un + 2v 2vn
zn + zn
∈ C ∀n ∈ N, z +1 =
n
2
(yn )
(zn )
z0
(un )n0
2u + 1 ∀n ∈ N, u +1 = 2u ∀n ∈ N, u +1 = 2+1
(xn )
lim zn
→+∞
n
R+
xn
(vn )
a
∈
(un ) (un ) un+1
(zn )
z0 = ρ
iθ
∀n ∈ N, z +1 = n
zn + zn
| |
2
n
R+
−u
n
u0 = a
∀n ∈ N, u +1 = n
k =0
uk
(un ) un =
− n+
(n
1) +
(un ) un+1 un n lim un
→+∞
n
··· +
n1
√ 2+ 1 +∞
− √n
u0 = 1
(un )
u0 = a
u0
0< a 0
∈ R ∀n ∈ N, u +1 = u
n
v0
n
∀ ∈ N, u +1 = 12
=
n u0 >
u0
∀n ∈ N, u +1 = 1 + ln u
(un )
u0
∈ R ∀n ∈ N, u +1 =
n
n
un
u0 > 0 n (un ) vn+1
−√ √aa
un un +
√a
n
un +
vn
| u −√√a | 2u0.v02 a
ln x + x = 0
2u0 .v02
n
→∞ 0
n
x> 0 α (un )
α (un )
u0 > 0
(un )
u0 = a (un )
∀n ∈ N, u +1 = 2+1 n
un
∈ [−2, 2] ∀n ∈ N, u +1 = √2 − u ∀n ∈ N, u ∈ [−2, 2] (u ) lim |u − 1| = 0 lim u n
n
n
n
( un
| − 1|)
n
n
1+
1+
√1 + · · · = 1 +
vn
n
−1
a un
n
n
un (un )
(un ) 2 n = un+1
∀n ∈ N, u +2u
(un )
∀n ∈ N, v
+1
(un )
n
n
2
n
|u | < 1
u0 = a > 0, u1 = b > 0 (un )
n
.
n
(un )
∀n ∈ N, u +1
1
∀n ∈ N, u +1 = 2 −u u
u0 = a
un un = o(n) (un )
(un )
(un )
a
1 1+
1 1+
√a
1/n
sin n1
+
m= 2 un N, n n1 , vn > m n max(n max(n0 , n1)
∃n1 ∈ ∀
→ < m
∃n0 ∈ N, ∀n n0, u
n
n
n 1 n+1
−
1 xnn ln xn n ln + =1
n
n
π
xn
2
− x → +∞
π
→2
(xn )
n
→ x0
R+
f n (xn+1) = x 1+1 [1, [1, + [ n
f (1/2) f (1/2) = xnn+1 + (xn ) = lim xn
···
x2 < 1 xn
x2
iθ
v0 =
−1
< 1
1
un
→ +∞
[1, [1, + [
R
x2
−x+1
∞
∆=
−3 < 0
= 2 + 1
uk
(un )
k=0
(un )
1 2
un+1
0
n
−
+
∞
(un ) f : x 1 + ln x [1, [1, + [ n 1 un+1 un = ln(u ln(un ) ln(u ln(un−1 ) (un ) u0 = 1 + ln u0 u0 g(x) = x 1 + ln x x g (x) = x1 1 0 ]1, ]1, + [ (un ) 1
→
∼u
g
n =1 un+1 =
=
un+1
u1
√ n → +∞
un un+1 =
+
−1
n
n
uk +
√a > 0
−1=
un
k=0
0=
n
n
uk =
k =0
u1 =
un u un+1 un = u +1 +u un+1 un = u +1 /1u +1
−
uk
k =0
(un )n1
||
−
− n
n 1 un+1
u0 = a, u1 = a2 , u2 = a4 a 0 (un ) (un ) =0 (un ) u0 > 0 + u0 < 0 (un )
− − −
−
n
−
−
un
n
− −
x
→ − 1
un vn = ln(u ln(un )
− u −1
∃
R
g
=
(un )
∞
−
∈
(un )
−1
f : x
(un )
|u +1 − √a| = 12 u | −√ n1 n
(un )
un
f : x
→
R+
1 2+
=
|u +1 − | = n
1 2+un
R+
− 2+1
vn+1 =
(un )
−1 + √2 − | 1 |u − | = (2+| )(2+ ) 4 |u − | = 41 |u0 − |
0
n
a
n
n
n
un
n
f : x
→
[−2, 2] [0, [0, 2] ⊂ [−2, 2] → √2 − x [0, 2] √ ∈ [0, [0, 2] →√ ∀n 1, u ∈ [0, = 2−u = 2− 2 + − 2 = 0
x un
n
un+1 n =1 = 2 0 =1 un+1 1 = 1+|u√2−−1u| un 1 α0 α> 0 1 + 2 un = |u|u+1−−1|1|
−
|
− |
n
| − 1|) √2 − u → 0 →1 u →1
| − |
n
√ − | u − 1| → 0
n
n
n
+
− | | −|| | | |un+1| 2|−|u a|| n
( un
n
|u | < 1 n
un
|u | |2−u |
|u | 2−|u | < 1
|u +1| |u | (|u |) |u | 2−|1 | |a| → 0 n
n
n
n
n
n
n
n
un u2 n
+
a
n
n
n =0 un < 1
| |
n
|u | |a| u →0 n
n
= + √ 2 (u − a) a = 2|u | =
1
1
−1
u1
+a n +a
=
n
2 au −2√ √ +2 au
n
√a| 2u v
−
f (un ) f (un )
= un
n
a un
√a −√ a
2
ln − ln1/ ++1 = (1−+1 1 2 C f (x) = + 1
un a
= vn2
− √a| n
vn = v02
R+
R
un+1 =
√1 + u
un )
un
un un un
un
x
f (x) =
x2
n =
√1 +
∈N
√
1+ 5 2
|u +1 − | = √1 + u − √1 +
−1
α
u0 = 1 0
n
1 2
−√a|
un
= 2u 2 u0 v02
f (u0 )f (u0 ) 0
→2
− u = 0
un+1 a
un un +
un
2
α = f −1 (0) (un )
f
u0 > 0
−√a
√ [ a, +∞[ √ = a
n
0 n
=
2
0
un
n
(un ) (un ) un
a
→ ln x + x
un+1 = un
ba
R+
1 2
n
−√
2
µ = ln a
[ a, + [
a x
x+
un a un
a
−√ √a = + a
n
un n0 un+1 = 2 unun un+1 2 unun
un+1 un+1
λ = ln ab (un )
a un
=
|u − √a| v |u
=
un un
un > 0 =0
n
√ ∞ → − √ | || | | − √ | √| √ √ | − | | − | → 1 2
un
1 2+x
n
b a
n
un
(un )
− 2v +1 + v
(vn ) (r 1)2 = 0 λ, µ R, vn = λn + µ v0 = ln a v1 = ln b n un = v = n ln +ln a = a ab
n
x g (0) = 0
vn+2
n
=
√1 +|uu +− √| 1 + |u 2− | n n
|u − | 21 |u0 − | n
n
n
n
un
→
1+
(vn ) (vn )
√1 + · · · =
1+
v0 = 1
|v +1 − | = n
n = 1 +
1
1 vn
− 1
vn
→
n
n
1 1+
1 1+
=
vn+1 = 1 +
1 vn
= n
n
> 1 1+
1
|v − | |v − | |v |
|v − | 1 |v0 − | n
∈N
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