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(xn)x0∈[a;b]
∀n∈N, xn+1 =1
2(f(xn) + xn)
f[a;b] [a;b]f
f[a;b]R
Zb
a
f(x) dx =Zb
a|f(x)|dx
f:x7→ Z2x
x
et
tdt
R∗
f
a b ab > 0
I(a, b) = Zb
a
1−x2
(1 + x2)√1 + x4dx
I(−b, −a)I(1/a, 1/b)I(1/a, a)I(a, b)
a, b > 1I(a, b)v=x+ 1/x
v= 1/t
a, b
ab > 0
fR
∀x∈R, f0(x) = f(2 −x)
+∞
un=Zn3
n2
dt
1 + t2∼1 n2
n∈N∗
(En): xn+x−1=0
(En)xn
limn→+∞xn= 1
yn= 1 −xnn
ln n
2n≤yn≤2ln n n
fn(y) = nln(1 −y)−ln(y)
ln(yn)∼ −ln n
xn= 1 −ln n
n+ o ln n n
n∈Nn≥2fR R
f(x) = xnsin 1
xx6= 0 f(0) = 0
fR
f
P∈R[X]≥2P0
x0P m ≥1
P0

a∈]0 ; π[n∈N∗C[X]R[X]
X2n−2 cos(na)Xn+ 1
(Pn)n∈N∗
P1=X−2∀n∈N∗, Pn+1 =P2
n−2
X2Pn
P(X)=(X−a)(X−b)∈C[X]
P(X)P(X3)
a=b P ∈R[X]a6=b a36=b3
R[X]
P a 6=b a3=b3
R[X]
u v E
|rg(u)−rg(v)| ≤ rg(u+v)≤rg(u) + rg(v)
u v L(R2)
rg(u+v)<rg(u) + rg(v)
u v R2
rg(u+v) = rg(u) + rg(v)
p∈Na∈R\ {0,1}Sp(un)
∃P∈Rp[X],∀n∈N, un+1 =aun+P(n)
u∈SpP Pu
SpR
φ u Pu
SpRk(X)=(X+ 1)k−aXk
k∈J0 ; pK
(un)
u0=−2un+1 = 2un−2n+ 7
f g E R C
f◦g= Id
ker(g◦f) = ker fIm(g◦f) = Img
E= ker f⊕Img
g=f−1
(g◦f)◦(g◦f)g◦f
f g E R C
f◦g= Id
ker(g◦f) = ker fIm(g◦f) = Img
E= ker f⊕Img
g=f−1
(g◦f)◦(g◦f)g◦f
E

f
ker f= Im f
f=u◦v
u v
EKn f g
E
h f g
rg f+ rg g−n≤rg(f◦g)
n= 3 E f2= 0
fRE3
f3+f= 0
x∈E x =y+z y ∈ker f z ∈ker(f2+ Id)
y=x+f2(x)z=−f2(x)
E= ker f⊕ker(f2+ Id)
dim ker(f2+ Id) ≥1x∈ker(f2+ Id) \ {0}
(x, f(x)) ker(f2+ Id)
det(−Id) dim ker(f2+ Id) = 2
E f
0 0 0
0 0 −1
0 1 0
fR3f3+f= 0
f
fR3= Im f⊕ker f
x∈E\ker ff(x), f2(x)
Im f f
a b
ax + 2by + 2z= 1
2x+aby + 2z=b
2x+ 2by +az = 1
A=
1−1 0 0
0 1 0 0
0 0 −1 1
0 0 0 −1
Ann∈Z
b= (i, j)B= (I, J)R
P b B
x∈E
v= Matbx V = MatBx
v V
f∈ L(E)
m= Matbf M = MatBf
m M
mn
f
a, b ∈R
ax +y+z+t= 1
x+ay +z+t=b
x+y+az +t=b2
x+y+z+at =b3
E n ≥2

E
E
(e1, . . . , en)E i ∈ {2, . . . , n}
(e1+ei, e2, . . . , en)E
E
E
E
E
fL(R3)
f3+f= 0
R3= ker f⊕Im f f
A=
000
001
0−1 0
A=
0111
1011
1101
1110
A B Mn(R)Mn(R)
X= tr(X)A+B
ERn > 1
f∈ L(E)
f∈ L(E)
L(E)
u, v :Rn[X]→Rn[X]
u(P) = P(X+ 1) v(P) = P(X−1)
rg(u−v)
(a, b)∈R2
Dn=
a+b b (0)
a
b
(0) a a +b
[n]
B=t(Com A)A∈ Mn(K)
rg A=n−1
C∈ Mn(K)
AC =CA =On
λ∈K
C=λB
a6=b λ1, λ2, . . . , λn
∆n(x) =
λ1+x a +x··· a+x
b+x λ2+x
a+x
b+x··· b+x λn+x
[n]
∆n(x)x
∆n(x) ∆n(0)

a b c
a2b2c2
a3b3c3
a+b b +c c +a
a2+b2b2+c2c2+a2
a3+b3b3+c3c3+a3
Dn=
1n n −1. . . 2
2 1 3
n−1 1 n
n n −1. . . 2 1
= (−1)n+1 (n+ 1)nn−1
2
n(xk) [0 ; π]
Pn=Y
1≤i<j≤n
(cos xi−cos xj)
Pn
(i, j)∈J1 ; 4K2M∈ M4(R)
mi,j = cos ((j−1)xi)
mi,j cos xi
det M P4|det M|<24
v n
S=
n
X
i=1 hv(ei), eii
(e1, . . . , en)E
T=
n
X
i=1
n
X
j=1 hv(ei), fji2
(e1, . . . , en) (f1, . . . , fn)E
T v r
(f|g) = R1
0f(t)g(t) dt
ERf1(x)=1 f2(x)=ex
f3(x) = x
a b f2(x)g(x) = ax +b
(A|B) = tr (AtB)Mn(R)
Sn(R)An(R)
M=
123
012
123
∈ M3(R)
S3(R)
H
Mn(R)
H J
E=C([−1 ; 1],R)
(f|g) = 1
2Z1
−1
f(x)g(x) dx
i∈ {0,1,2,3}Pi(x) = xi