énoncé

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(fn)n∈N∗]−1,+∞[
fn(x) = xnln(1 + x).
fn
n∈N∗hn]−1,+∞[
hn(x) = nln(1 + x) + x
1 + x.
hn
hn(0) hn
n= 1
f1]−1,+∞[f0
1(x)h1(x)
f1]−1,+∞[
n∈N∗\ {1}
fn]−1,+∞[f0
n(x)hn(x)
fn]−1,+∞[n n
(Un)n∈N∗
Un=
1
Z
0
fn(x)dx.
U1
a b c
∀x∈[0,1],x2
x+ 1 =ax +b+c x+ 1.
1
Z
0
x2
x+ 1 dx.
U1=1
4

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(Un)n∈N∗
(Un)n∈N∗
(Un)n∈N∗
∀n∈N∗,06Un6ln 2
n+ 1.
(Un)n∈N∗
Unn>2
x∈[0,1] n∈N∗\ {1}
Sn(x) = 1 −x+x2+· · · + (−1)nxn=
n
X
k=0
(−1)kxk.
Sn(x) = 1
1 + x+(−1)nxn+1 1 + x.
n
X
k=0
(−1)k
k+ 1 = ln 2 + (−1)n
1
Z
0
xn+1
1 + xdx.
Un
Un=ln 2
n+ 1 +(−1)n n+ 1 ln 2 −1−1 2+· · · +(−1)k k+ 1 +· · · +(−1)n n+ 1 .
A=
013
013
004
D=
000
010
004
A
λ A −λI
λ(A−λI)X= 0
X∈ M3,1(R)
P
λ= 0
λ= 1 λ= 4
P P −1A=P DP −1
M2 =A
(1) : M2=A M
M N =P−1M P P
M2=A⇐⇒ N2=D
N2=D N D =D N
N2=D N
N N2=D
B(1)

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Q
Q(0) = 0, Q (1) = 1, Q (4) = 2.
−1
6A2+7 6A=B B
F
A F =F A ⇐⇒ B F =F B.
n2p]0; 1[ q= 1 −p
p q
•
•n
k PkFk
k
TnXnYn
TnXnYn
(Ω; A;P)
Tn
k[[1; n−1]] k= 1 P(Tn=k)
P(Tn=n)
n
X
k=1
P(Tn=k)=1
TnE(Tn) = 1−qn
1−q
Xn
Xn
E(Xn)=1−qn
Yn
k[[0; n−1]] P(Yn=k)
P(Yn=n)
TnXnYnE(Yn)
(Tn)n∈N∗T
TnXnYn

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