Devoir maison 10

∀q∈N∗,X
k≥qk
qxk−q
∀x∈]−1,1[,
+∞
X
k=qk
qxk−q=1 (1 −x)q+1
p∈]0,1[ r∈N∗
t= 0
t= 1
p
r
X
X
X
n n
n n
p p ∈]0,1[
x
X
n−X
Y
i∈[|0, n|]k∈NP(Y=k|X=i)
Z=X+Y
Z
A, B C
t= 0 A
n∈N
AnA n

BnB n
CnC n
P(An) = anP(Bn) = bnP(Cn) = cn
an+1 anbncn
bn+1 cn+1 anbncn
A=
01
2
1
2
1
201 2
1
2
1
20
A
P D D =P−1AP
P−1
an, bn
cnn
100 1
2
n∈N∗
n n
pn
lim
n→+∞pn
U1U2
U1
U2
U1
U2
∀n∈N∗Bnn
∀n∈N∗, pn=P(Bn)
p1
∀n∈N∗, pn+1 =−6
35pn+4 7
n pn