Enoncé

M2(R)
A=1 0
0 0, B =0 1 0 0, C =0 0 0 1, T =1 1 0 1
E=a b
0c,(a, b, c)∈R3
E(A, B, C)E
E
∀(M, N)∈ E2, MN ∈ E
MEM M−1∈ E
Ef(M) = T MT
fE
T f E
T
F f (A, B, C)E
f(A), f(B), f (C) (A, B, C)F
f
f
f
λ f(M) = λM M ∈ E
I=
100
010
001
H=
000
101
000
H2aRnN(I+aH)n
nNFn
GM3(R)G3=F g E
g◦g◦g=f

f
∀x∈R, f (x) = 1
√1 + x2
(un)
u0=
1
R
0
f(x)dx
∀n∈N∗, un=
1
R
0
xnf(x)dx
Cff, O,~ i,~ j.
f.
fR
f[0,+∞[
f x +∞.
fR
Cf
f[0,+∞[J
y]0,1] , x [0,+∞[
f(x) = y
f−1
F x
F(x) = ln x+px2+ 1
λA(λ)
M(x, y)
λ6x62λ06y6f(x)
A(λ) = Z2λ
λ
f(x)dx
∀x∈R, x +px2+ 1 >0
F.
F f R
F
F x +∞. F x
−∞
A(λ)λA(λ)λ+∞.

(un).
u0u1.
u3.
x3
√1 + x2=x2x √1 + x2
(un).
(un)
∀x∈[0,1] ,∀n∈N,06xn
√1 + x26xn
∀n∈N×,06un61
n+ 1
(un)
P(B/A)B A
P(B/A) = PA(B)
A B
A60% B40%
A0.1
B0.2

f1/f2)
A
Y λ = 20.
X
A
Y Y
Y
k n P [X=k/Y =n]
k6n k > n
(Y=i)i∈N, X
fR
f(t) = 2
(1 + t)3t>0
f(t) = 0 t < 0
f Z
FZZ
+∞
Z
0
2t
(1 + t)3dt
u=t+ 1
Z
Z
A B Z1Z2Z1Z2
Z
C=B
D=B
P(C), P (D), P (D/C).

T= max(Z1, Z2)GTT
(T6x) (Z16x) (Z26x)
∀x∈R, GT(x)=[FZ(x)]2
T
A
B
A M
B N
[0,1]
M N
v M w N
S
S M N