HEC Eco 2004

\
(cn)n∈N×n
cn=
1
Z
0
xn−1
1 + xdx
(cn)n∈N×
n cn+1 +cn=1
n
n2
1
n62cn61 n−1
cnn
c1n2
cn= (−1)n n−1
X
k=1
(−1)k+1
k−ln 2!
n
cn
n fnR R
fn(t) = 
0t < 1
1
cntn(1 + t)t>1
n
x1
x
Z
1
1
tn(1 + t)dt =
1
Z
1/x
un−1
1 + udu
n fn
(Xn)n∈N×
(Ω,A,P)n Xn
[1,+∞[fnFn
Xn
n Xn
Xncncn−1
n1F1
yP([X16y]) >1
2
Z= ln(X1)
x1
06
1
Z
1/x
un
(1 + u)2du 61 n+ 1
lim
n→+∞
1
Z
1/x
un
(1 + u)2du
.

\
n Fn(x)
lim
n→+∞Fn(x) = 1.
lim
n→+∞Fn(x)x1
(Xn)n∈N×
n E
2n
k Xkx7→ xk
(1, X, . . . , X2n)E
a0, a1, . . . , a2n2n+ 1 QR
Q(x) =
2n
X
k=0
akxk,
s(Q)
s(Q)(x) =
2n
X
k=0
a2n−kxk.
s(Q)Q
n2Q(x) = 4x4+ 7x3+ 2x2+ 1 s(Q)(x) = x4+ 2x2+ 7x+ 4
s
s Q 7→ s(Q)E
M=
001
010
100
M M
n= 1 M s
(1, X, X2)s
(A0, ...., A2n)
x, 
Ak(x) = x2n−k+xk06k6n−1
An(x) = xn
Ak(x) = xk−x2n−kn+ 1 6k62n
s◦s
P λ s(P) = λP
s◦s(P)s{1,−1}
s(Ak)k06k62n
(A0, ...., A2n)
s

\
(Rk)k∈N×
x,
R1(x) = x, R2(x) = x2−2
k2,
Rk+1(x) = xRk(x)−Rk−1(x)
R3R4
k Rkk
x
Rkx+1
x=xk+1 xk
a x
x+1
x=a
s1
Q2n Q(x) =
2n
P
k=0
akxka2n
k[[0, n]] ak=a2n−k
e
Q
e
Q(x) = an+
n
X
k=1
an−kRk(x).
0Q
x y =x+1
x
Q(x)
xne
Q(y)
Q
n3Q
Q(x) = x6+x5−9x4+ 2x3−9x2+x+ 1.
Q
p2
ΩE[[1, p]]
AΩPA
QΩ
P({Q}) = 1
card(Ω)
QΩi Q s(Q)i
i k ak=a2n−k
Z Q Ω
Q s(Q)
n= 2 Q(x) = x4+ 7x3+ 2x2+ 5x+ 1 Z(Q)=3
n1p2
ΩZ
Z
n p 2
Ω

\
Z
P([Z= 1]) = p−1
pn
Z2n+ 1
P([Z= 2n+ 1]) = 1
pn
Z j 06j6n
P([Z= 2j+ 1])
Y=Z−1
2. Y
Z n p