Exercices de Colles de Sup

2012 −2013 2013 −2014

A B C E
A∪B=A∪C A ∩B=A∩C B =C
E F f :E−→ FΦ : P(E)−→
P(F) Φ(A) = fhAi
Φf
f f(A) = f(B)A=B
Φ
E F f :E−→ F A B ∈P(E)
f(A∩B)⊂f(A)∩f(B)
f
A B f
E f, g :E−→ E
f g f ◦g
E=NE g(n) = n+ 1 f(n) = n n ≥1
f(0) = 1

E f :E−→ E f ◦f◦f=f
f f
f f ◦f=Id f
f x x =f(y)f(f(x)) = x
f
E F
E F F
E
i:E−→ F f(i(x)) = x x ∈E
s:F−→ E g(x)x s
N N2
n > 0N Nn
N
R R\N
f(n) = n+1
2n∈Nf(n+1 m) =
n+1
m+1 m, n ∈Nm≥2f(x) = x
E E
f E
E
E x ∈E A ={fn(x)|n∈N}A6=E
A fn(x)6=x n A\{x}
σN N
n σ(n)≥n
N σ(n)σ(n)≥n
n∈[[0, N]] σ(n)∈[[0, N]] σ[[0, N]]
σ(N+ 1) ≤N
E f :P(E)−→ P(E)
A⊂B f(A)⊂f(B)f
A A ⊂f(A)
f(A) = A
n∈N∗Pn
k=0
k
(k+1)!
1−1
(n+1)!

Pn
k=1 1
n+k
un+1 −un
n∈N∗k
n
kn
k+1 n kp
n= 2p p p + 1 n= 2p+ 1
(x, y, z)∈R3
x3y4z= 2
xyz4= 4
x5yz2= 16
ln |x|ln |y|ln |z| |x|=
|y|= 1 |z|= 2 x y z
(1,1,2) (−1,−1,−2)
n∈N∗Pn
k=1
k
2k
n→ ∞
k=Pk
i=1 1 2
qk
2−1
2n−1−n 2n2
E n ∈N∗
X
A,B⊂E|A∩B|
n4n−1A
B|A∩B| |B|
A|B|2n−1
m n k N∗
m+n
k=X
p+q=km
pn q
xk(1 + x)m+n
(1 + x)m(1 + x)n
n=pα1
1...pαk
k∈N∗pi
n
pβi
iβi≤αi
pαi
i−1
pi−1
nd(n)
2d(n) = (α1+
1)...(αk+ 1)

(pn)
Pn
k=1 1
pk+∞n→ ∞
Pn=Qn
k=1 1
1−1
pk∞
Pn≥1 + ... +1
nln 1
1−1
pk
k1
2pk
z+ ¯z=z4
z z4∈Rz
a(1 + i)a(1 −i)a∈R
03
√2−1+i
3
√2−1−i
3
√2
u z Cu6= 1 z /∈R
z−u¯z
1−u∈R|u|= 1
θ∈Rn∈N∗
Pn−1
k=0 cos3kθ
3
4cot π 2n−1 4cot 3π 2n
a b c d a +c=b+d
a+ib =c+id
a b c d
H={z∈C|=(z)>0}z∈H
f(z) = z−i
z+iDf f H
D
f z0−→ iz0+1
z0−1f−1(z0)∈H
|z0|<1
f