Cours «Suites Usuelles

α r
α r u
(u0=α
∀n∈Nun+1 =r+un
r∈Cu r
∀n∈N, un=nr +u0
∀n∈N
n
X
k=0
uk= (n+ 1)u0+un
2
r∈Ru
r
r > 0u−−→
+∞+∞
r= 0 u u0u−−→
+∞u0
r < 0u−−→
+∞−∞
α r
α r u
(u0=α
∀n∈Nun+1 =r.un
r∈Cu r
∀n∈N, un=rnu0
r6= 1
∀n∈N
n
X
k=0
uk=u0
1−rn+1
1−r

r= 1
∀n∈N
n
X
k=0
uk= (n+ 1)u0
r∈Ru r
r∈]− ∞,−1] u
r∈]−1,1[ |r|<1u
r= 1 u u0
u0
r∈]1,+∞[u+∞u0>0−∞
r r ∈]−1,1] |r|<1
r= 1
u
a b ∀n∈Nun+1 =aun+b
a= 1 u
b= 0 u
a b
u
∀n∈Nun+1 =aun+b(E)
u v (E)u−v
un−vnn∈N
a
v(E)
u a u +v
un+vnn∈N(E)
v(E)u∈CN
u(E)u−v
a
v(E)u
(E)u−v a
u−v
a v + (u−v) (E)
v+ (u−v) = u

a6= 1
(E)
(E)
a= 1 (E)b
u∈CNu E
∀n∈Nun=u0+nb
a6= 1
α
α=aα +b
(E)u∈CN
u−α(un−α)n∈Na
(E)u∈CN
∀n∈Nun=α+an(u0−α)
u
(u0= 0
∀n∈Nun+1 = 2un+ 1
10 000 e18% 5 60
1,5%
∀n∈Nun+2 =aun+1 +bun(E)
a b u
r2−ar −b= 0 (C)
a b
a b b 6= 0
∀n∈Nun+2 =aun+1 +bun(E)
(C) (E) ∆
(C)

∆>0 (C)r1r2
(E) (λrn
1+µrn 2)n∈N
λ µ
∆ = 0 (C)r0
(E) (λrn
0+µnrn 0)n∈N
λ µ
∆<0 (C)reiθ
re−iθ (E)
(rn(λcos(nθ) + µsin(nθ)))n∈Nλ µ
(E)u0u1
u0u1
λ µ u0u1
a b
b6= 0 u∀n∈Nun+2 =aun+1 +bun
u0u1
u0u1
(un)n∈N
u0= 0 u1= 1
∀n∈Nun+2 =un+1 +un
unn
u∀n∈N, un+1 =f(un)
fDR R
DRf:D → Ra∈R
u
(u0=a
∀n∈Nun+1 =f(un)

a /∈ D
u1
f x 7→ √x−1a9a∈R+
u1= 2 u2=√2−1<1u3=p√2−1−1<0u4
ADARf
f∀x∈A f(x)∈A
a∈A
n∈NP(n)unun∈A
n∈Nun
u
a∈[−1,+∞[u
(u0=a
∀n∈Nun+1 =√1 + un
f x 7→ √1 + x[−1,+∞[
f(un)n∈N
nNun
[−1,+∞[
v
(v0= 5
∀n∈Nvn+1 =v
3
2
n−1
f:R+→R
x7→ x3
2−1
fR+f
f(0) /∈R+A= [4,+∞[
A f f
fR+f(4) = 7 ≥4x∈A
f(x)≥f(4) ≥4f(x)∈A5A
v
ARf
A f(A)⊂A
(un)n∈Nl f
l f(l) = l l f
(un)n∈N