# 1 Introduction sur les suites numériques

U A IN IR
U:AIR
n7→ UnAIN n U Un
U n 2nU:n7→ 2n
× ×
×
×
0 1 2 3
1
4
9
U(Un)nA(Un)Un
(Un)UnnA
U0= 20= 1 Uk= 2k(k+ 1)e
k+ 1
(Vn)n1Vn=1
nV1= 1 Vk=1
k
kek
(Un)
nA Un+1 Un
nA Un+1 > Un
nA Un+1 Un
nA Un+1 < Un
(Un)
(Un)
nA Un+1 =Un
• ∃n0A, nA, n n0Un=Un0
(2n) ( 1
n) (E[4 + 5
n+1 ])
U0= 9 U1= 6 U2=U3=U4= 5 n55
n+1 <5
6<1n5, 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) <0n1
(n2+n)
((n+ 1)2+ (n+ 1)) (n2+n) = n2+ 2n+1+n+ 1 n2n= 2n+ 2 = 2(n+ 1) >0n0
(Un)Un+1
Un
(2n)n0,2n>02n+1
2n= 2 >1
(2n
n!)nINn1,2n
n!>02n+1
(n+1)!
n!
2n=2
n+1 1n+ 1 2
(Un)Un=f(n)f(Un)
f
(nln n)n1
f(x) = xln x x [1,+[f[1,+[f0(x)=11
x0
x10<1
x1f[1,+[ (nln n)n1
Un+1 =f(Un) (Un)f
f(Un)
Un=f(n)f(x) = sin(2πx)xIR+
fnIN, Un= sin(2πn) = 0 (Un)
(Un)Un+1 =2Un+ 1,n
U0= 3 (Un)
(Hn)Un+1 Un0n
U1U0=23+13 = 790 (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)
• ∃MIR, nA, UnM
• ∃mIR, nA, Unm
• ∃MIR+,nA, |Un| ≤ M
(sin n)nIN, |sin n| ≤ 1
(n(1)n)
(n(1)n)
MIR, nIN, n(1)nM
n n = 2k k IN (2k)(1)2k= 2kM
n
(ln n
n)n>0
f:x7→ ln x
x[1,+[
f[1,+[f0(x) = 1/x xln x
x2=1ln x
x2
f0(x)01ln xex 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,+[,0f(x)1
enIN,0f(n)1
e
(ln n
n)n>0
(Un)Un+1 =2Un+ 35,n
U0= 0
0Un7n(Hn)
U0= 3 [0,7] (H0)
Un+1 =2Un+ 35
Un0 2Un+ 35 0Un+1
Un72Un+ 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,NIN, n>N :|Un`|< ε
(Un)`lim
n→∞ Un=` Un`
×××××××××××
`
`+ε
`ε
N
(1
n) 0
ε > 0,NεIN, n>Nε,1
n< ε n > 1
εNε=E(1
ε)+1>1
ε
((1)n)
((1)n)`ε > 0,NIN, 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,|ab|< ε
ε > 0 lim
n→∞ Un=a⇔ ∃N1IN, n>N1:|Una|<ε
2
lim
n→∞ Un=b⇔ ∃N2IN, n>N2:|Unb|<ε
2
N= max(N1, N2)n>N:|ab|=|(aUn)+(Unb)| ≤ |aUn|+|Unb|< ε
(Un) (Vn)
UnVnlim
n→∞ Unlim
n→∞ Vn.
Un=sin n
n0Vn=1
n+ 1 1nIN, UnVnlim
n→∞ Unlim
n→∞ Vn
P(n)
NIN, n>N :P(n)
Un< Vnlim
n→∞ Un<lim
n→∞ Vn
Un< Vnlim
n→∞ Unlim
n→∞ Vn
Un= 0 Vn=1
nnIN, Un< Vnlim
n→∞ Un= lim
n→∞ Vn= 0
(Un) (Vn)
UnMlim
n→∞ UnM
Vnmlim
n→∞ Vnm
N1,n>N1, UnVn
lim
n→∞ Un=a⇔ ∀ε > 0,N2IN, n>N2,|Una|< ε Un> a ε
lim
n→∞ Vn=b⇔ ∀ε > 0,N3IN, n>N3,|Vnb|< ε Vn< b +ε
ab⇔ ∀ε > 0, a < b +ε
ε > 0N=max(N1, N2, N3)n>N a<Un+εVn+ε<b+ 2ε
(1
n) 0 n1,|1
n| ≤ 1 ( 1
n)
((1)n))
lim
n→∞ Un=`⇔ ∀ε > 0,NIN, n > N, |Un`|< ε
ε= 1 n>N |Un`|<1
|Un|=|Un`+`| ≤ |Un`|+|`| ∀n>N |Un|<1 + |`|
M= max{U0, U1,··· , UN,1 + |`|} ∀nIN |Un| ≤ M(Un)
(Un) (Vn) (Wn)
UnVnWn(Un) (Wn)`(Vn)
(Vn)n1,3 + 2
nVn3n+1
n
lim
n+Vn= 3
N1,n>N1, UnVnWn
lim
n→∞ Un=`⇔ ∀ε > 0,N2IN, n>N2,|Un`|< ε Un> ` ε
lim
n→∞ Wn=`⇔ ∀ε > 0,N3IN, n>N3,|Wn`|< ε Wn< ` +ε
lim
n+Vn=`⇔ ∀ε > 0,NIN, n > N, ` ε<Vn< ` +ε
ε > 0N=max(N1, N2, N3)n > N ` ε<UnVnWn< ` +ε
|Vn| ≤ Unlim
n+Un= 0 lim
n+Vn= 0
(sin n
n) 0 nIN,|sin n
n| ≤ 1
n
1
n0
(Un)
(Un) lim
n+Un= sup{Un, n IN}inf{Un, n IN }
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# 1 Introduction sur les suites numériques

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