1 Introduction sur les suites numériques

U A IN IR
U:A→IR
n7→ UnA⊂IN n U Un
U n 2nU:n7→ 2n
× ×
×
×
0 1 2 3
1
4
9
U(Un)n∈A(Un)Un
(Un)Unn∈A
U0= 20= 1 Uk= 2k(k+ 1)e
k+ 1
(Vn)n≥1Vn=1
nV1= 1 Vk=1 k
kek
(Un)
•n∈A Un+1 ≥Un
•n∈A Un+1 > Un
•n∈A Un+1 ≤Un
•n∈A Un+1 < Un
•(Un)
•(Un)
•n∈A Un+1 =Un
• ∃n0∈A, ∀n∈A, n ≥n0⇒Un=Un0

(2n) ( 1
n) (E[4 + 5 n+1 ])
U0= 9 U1= 6 U2=U3=U4= 5 n≥55
n+1 <5 6<1∀n≥5, Un=E[4+ 5 n+1 ]=4
(Un+1 −Un)
(1
n)n>01 n+1 −1 n=n−(n+1) n(n+1) =−1 n(n+1) <0∀n≥1
(n2+n)
((n+ 1)2+ (n+ 1)) −(n2+n) = n2+ 2n+1+n+ 1 −n2−n= 2n+ 2 = 2(n+ 1) >0∀n≥0
(Un)Un+1
Un
(2n)∀n≥0,2n>02n+1
2n= 2 >1
(2n
n!)n∈IN∗∀n≥1,2n n!>02n+1 (n+1)!
n!
2n=2 n+1 ≤1n+ 1 ≥2
(Un)Un=f(n)f(Un)
f
(n−ln n)n≥1
f(x) = x−ln x x ∈[1,+∞[f[1,+∞[f0(x)=1−1
x≥0
x≥1⇒0<1
x≤1f[1,+∞[ (n−ln n)n≥1
Un+1 =f(Un) (Un)f
f(Un)
Un=f(n)f(x) = sin(2πx)x∈IR+
f∀n∈IN, Un= sin(2πn) = 0 (Un)
(Un)Un+1 =√2Un+ 1,∀n
U0= 3 (Un)
(Hn)Un+1 −Un≤0∀n
U1−U0=√2∗3+1−3 = √7−√9≤0 (H0)
Un+2 −Un+1 =p2Un+1 + 1 −√2Un+ 1 = (p2Un+1 + 1 −√2Un+ 1)√2Un+1+1+√2Un+1
√2Un+1+1+√2Un+1
=(2Un+1+1)−(2Un+1)
√2Un+1+1+√2Un+1 =2(
≤0Hn
z }| {
Un+1 −Un)
p2Un+1 +1+p2Un+ 1
| {z }
≥0
≤0 (Hn+1)
(Hn)n(Un)
(Un)
• ∃M∈IR, ∀n∈A, Un≤M
• ∃m∈IR, ∀n∈A, Un≥m
• ∃M∈IR+,∀n∈A, |Un| ≤ M

•(sin n)∀n∈IN, |sin n| ≤ 1
•(n(−1)n)
(n(−1)n)
∃M∈IR, ∀n∈IN, n(−1)n≤M
n n = 2k k ∈IN (2k)(−1)2k= 2k≤M
n
•(ln n
n)n>0
f:x7→ ln x
x[1,+∞[
f[1,+∞[f0(x) = 1/x x−ln x
x2=1−ln x x2
f0(x)≥0⇔1≥ln x⇔e≥x f(1) = 0 f(e)1
elim
x→+∞
ln x
x= 0
x
f0(x)
f(x)
1e+∞
+0−
00
1
e
1
e
00
∀x∈[1,+∞[,0≤f(x)≤1
e∀n∈IN∗,0≤f(n)≤1 e
(ln n
n)n>0
•(Un)Un+1 =√2Un+ 35,∀n
U0= 0
0≤Un≤7∀n(Hn)
U0= 3 ∈[0,7] (H0)
Un+1 =√2Un+ 35
Un≥0 2Un+ 35 ≥0Un+1
Un≤7⇒2Un+ 35 ≤49 ⇒Un+1 ≤√49 ⇒Un+1 ≤7
(Hn+1)
(Hn)n(Un)
(Un) (Vn)Un+Vn
UnVn
Un=n Vn= 2n+ 1 Wn= ln(ch n +sh n)
Un+Vn= 3n+ 1 UnVn=n(2n+ 1) Un=Wn

(Un)`∈IR
∀ε > 0,∃N∈IN, ∀n>N :|Un−`|< ε
(Un)`lim
n→∞ Un=` Un→`
×××××××××××
`
`+ε
`−ε
N
(1
n) 0
∀ε > 0,∃Nε∈IN, ∀n>Nε,1
n< ε ⇔n > 1 εNε=E(1 ε)+1>1 ε
((−1)n)
((−1)n)`∀ε > 0,∃N∈IN, ∀n > N, |(−1)n−`|< ε
ε=1
2n|(−1)n−`|=|1−`|<1 2|(−1)n+1 −`|=| − 1−`|<1 2
n>N 2 = |1−(−1)|=|(1 −`)−(−1−`)|≤|1−`|+| − 1−`|<1
(Un)
lim
n→∞ Un=alim n→∞ Un=b a =b⇔ ∀ε > 0,|a−b|< ε
ε > 0 lim
n→∞ Un=a⇔ ∃N1∈IN, ∀n>N1:|Un−a|<ε
2
lim
n→∞ Un=b⇔ ∃N2∈IN, ∀n>N2:|Un−b|<ε
2
N= max(N1, N2)∀n>N:|a−b|=|(a−Un)+(Un−b)| ≤ |a−Un|+|Un−b|< ε
(Un) (Vn)
Un≤Vnlim
n→∞ Un≤lim n→∞ Vn.
Un=sin n
n→0Vn=1 n+ 1 →1∀n∈IN∗, Un≤Vnlim
n→∞ Un≤lim n→∞ Vn
P(n)
∃N∈IN, ∀n>N :P(n)
Un< Vnlim
n→∞ Un<lim n→∞ Vn
Un< Vn⇒lim
n→∞ Un≤lim n→∞ Vn
Un= 0 Vn=1
n∀n∈IN∗, Un< Vnlim
n→∞ Un= lim n→∞ Vn= 0

(Un) (Vn)
Un≤M⇒lim
n→∞ Un≤M
Vn≥m⇒lim
n→∞ Vn≥m
∃N1,∀n>N1, Un≤Vn
lim
n→∞ Un=a⇔ ∀ε > 0,∃N2∈IN, ∀n>N2,|Un−a|< ε Un> a −ε
lim
n→∞ Vn=b⇔ ∀ε > 0,∃N3∈IN, ∀n>N3,|Vn−b|< ε Vn< b +ε
a≤b⇔ ∀ε > 0, a < b +ε
ε > 0N=max(N1, N2, N3)∀n>N a<Un+ε≤Vn+ε<b+ 2ε
(1
√n) 0 ∀n≥1,|1 √n| ≤ 1 ( 1 √n)
((−1)n))
lim
n→∞ Un=`⇔ ∀ε > 0,∃N∈IN, ∀n > N, |Un−`|< ε
ε= 1 ∀n>N |Un−`|<1
|Un|=|Un−`+`| ≤ |Un−`|+|`| ∀n>N |Un|<1 + |`|
M= max{U0, U1,··· , UN,1 + |`|} ∀n∈IN |Un| ≤ M(Un)
(Un) (Vn) (Wn)
Un≤Vn≤Wn(Un) (Wn)`(Vn)
(Vn)∀n≥1,3 + 2
n≤Vn≤3n+1 n
lim
n→+∞Vn= 3
∃N1,∀n>N1, Un≤Vn≤Wn
lim
n→∞ Un=`⇔ ∀ε > 0,∃N2∈IN, ∀n>N2,|Un−`|< ε Un> ` −ε
lim
n→∞ Wn=`⇔ ∀ε > 0,∃N3∈IN, ∀n>N3,|Wn−`|< ε Wn< ` +ε
lim
n→+∞Vn=`⇔ ∀ε > 0,∃N∈IN, ∀n > N, ` −ε<Vn< ` +ε
ε > 0N=max(N1, N2, N3)∀n > N ` −ε<Un≤Vn≤Wn< ` +ε
|Vn| ≤ Unlim
n→+∞Un= 0 lim n→+∞Vn= 0
(sin n
n) 0 ∀n∈IN∗,|sin n n| ≤ 1 n
1
n→0
(Un)
(Un) lim
n→+∞Un= sup{Un, n ∈IN}inf{Un, n ∈IN }