Suites homographiques : Devoir de maths MPSI

2015/2016
fR\−d
c∀x∈R\−d c, f(x) = ax +b cx +da, b, c d
c(ad −bc)6= 0
f(x) = x1 2 R\−d
c
f
(un)n∈Nu0∈R\−d
c∀n∈N, un+1 =f(un)
u0
f(un)n∈N
f α p up=α
(un)n∈Np > 0up−1=α
(un)n∈N
f α β u06=α
(vn)n∈N∀n∈N, vn=un−β
un−α
(vn)n∈N
cα +d
cβ +d
vnn u0unn u0
unu0
(un)n∈Nu0/∈E E
f α u06=α(vn)n∈N
∀n∈N, vn=1
un−α
(vn)n∈N
vnn u0unn u0
unu0
(un)n∈Nu0/∈E E
(un)n∈Nu0= 2 ∀n∈N, un+1 =4un+ 2
un+ 5
∀n∈N, un>0
4x+ 2
x+ 5 =x α β α < 0β > 0
(vn)n∈N∀n∈N, vn=un−β
un−αvnn

2015/2016
(un)n∈N
(un)n∈Nu0= 4 ∀n∈N, un+1 =7un−12
3un−5
∀n∈N, un>2
7x−12
3x−5=x α
(vn)n∈N∀n∈N, vn=1
un−αvnn
(un)n∈N