Lycée Jean Perrin Classe de TSI Révisions d'oral (4) Sujet 1

TSI2
I=Z+∞
0
t
t2+ 4t+ 3
I
(a, b)∈R21
t2+ 4t+ 3 =a t+ 1 +b t+ 3
I
J=Z+∞
0
t
t2+t+ 1
A=
3 1 . . . 1
1 1 . . . 1
1 1 . . . 1
∈ Mn(R)n>3
AMn(R)
A
0n−2
α β A α +β=n+ 2
Tr(A2)α β
E=R2[X]φ:E×E→R
(P, Q)7→ P(−1)Q(−1) + P0(0)Q0(0) + P00(0)Q00 (0)
φ E
E
X2R1[X]
pi,n
πnπn= (p1,n, p2,n, p3,n, p4,n)
πn+1 πnπn+2 πn
(p2,2n)n∈N(p4,2n)n∈Np1,2np3,2n
n+∞pi,2npi,2n+1

TSI2
An
Bn
n > 3P(An+1 ∩An)P(An)P(An+1 ∩An)P(An−2)
(An)
n > 3P(An+1) = P(An) + 1
8(1 −P(An−2)
P(An)n∈N
AnBk
∀n>2un=
n
X
k=2
ln k
k
t7→ ln t
t
∀k>4,Zk+1
k
ln t
tt6ln k k6Zk
k−1
ln t
tt
ununn
Xun
n2