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θ∈R2π
Sn=
n
X
k=0
cos(kθ)un=cos(nθ)
n
(Sn)n∈N
cos(nθ) = Sn−Sn−1
un
|cos x| ≥ cos2x
|un|
un= cos n2πln(1 −1/n)
X
n≥2
1
nln n
x∈]0 ; 1[ ∪]1 ; +∞[
f(x) = Zx2
x
dt
ln t
f(x)x∈]0 ; 1[ ∪]1 ; +∞[
x > 1
Zx2
x
xdt
tln t≤f(x)≤Zx2
x
x2dt
tln t
f1+
f1−
fC1]0 ; 1[ ]1 ; +∞[
f0(x)
fC1
C∞]0 ; +∞[
Z+∞
0
sin t
et−1dt=
+∞
X
n=1
1
n2+ 1
2π
f(t) = ch t t ∈[−π;π]
Zπ
0
ch t. cos(nt) dt= (−1)nsh π
n2+ 1
n∈N∗
un=Z+∞
0
t−[t]
t(t+n)dt
[t]t
(un)n≥1
A > 0
ZA
0
t−[t]
t(t+n)dt=1 n Zn
0
t−[t]
tdt−ZA+n
A
t−[t]
tdt!
un
vn=nun
vn−vn−1−1
2n

un
Z+∞
0
sin t
tdt
x
f(x) = Z+∞
x
sin t
tdt
fC1R
Z+∞
0
f(t) dt
I=Z+∞
0
sin3t
t2dt
x > 0
I(x) = Z+∞
x
sin3t
t2dt
sin 3a= 3 sin a−4 sin3a
I(x) = 3
4Z3x
x
sin t
t2dt
I
I=Z+∞
0
ln 1 + t2
t2dt
u= tan t x ∈R
Zx
0
dt
3 + cos2t
x7→ Z2x
x
dt
√1 + t2+t4
x→0x→ ±∞
x∈[1 ; +∞[
F(x) = Zx
1
t
√t3−1dt
F[1 ; +∞[
C∞]1 ; +∞[
F0(x)
F1F1
F+∞
F[1 ; +∞[
F−1]0 ; +∞[
yy0=py3−1
F−10
un=r(1) + r(2) + ··· +r(n)
n2
r(k)n k
vn=(n−r(1)) + (n−r(2)) + ··· + (n−r(n))
n2

x > 0
F(x) =
+∞
X
n=0
(−1)n
n+x
F
FC1C∞
F(x) + F(x+ 1)
x > 0
F(x) = Z1
0
tx−1
1 + tdt
F+∞
f(x) =
+∞
X
n=1
1
n2+x2
fC1R
f
+∞f
+∞
X
n=1
1
n2=π2 6
+∞
X
n=1
1
n4=π4 90
C
u, v CE
u◦v=v◦u
u v
MMn(R)
tr M= 0 M3−4M2+ 4M=On
E=CNf:E→E u = (un)
v= (vn)
v0=u0∀n∈N∗, vn=un+un−1
2
f
EKn∈Np∈ L(E)p2
p
p p3=p
E E
F:L(E)→ L(E)
F:f7→ 1
2(f◦p+p◦f)
F
F
A∈ Mn(R)
A3+In=On
A

A∈ Mn(R)
A
A=
3 6 −5−2
−1−6 5 −2
−1−10 8 −3
0−3 2 0
B=
12621
0 2 2 5
0 0 3 2
0 0 0 5
B, C ∈ Mn(C)
x∈CxIn−B xIn−C
(xIn−B)−1(xIn−C)−1
A∈ Mn(C)PA(x) = det(xIn−A)P0
A
PA
x A
tr (xIn−A)−1=P0
A(x)
PA(x)
M=A A
(0) A
A∈ Mn(R)
∀P∈R[X], P (M) = P(A)AP 0(A)
(0) P(A)
M
M∈ Mn(C)M5=M2tr(M) = n
EKF, G
p F
G s F G
f F
φ(f) = p◦f◦s
φL(E)
φ φ
φ
p s
E=F⊕G
I=ZZD (1 + xy) dxdy
D O
f: [0 ; 1] →[0 ; 1]
f◦f=f
{x∈[0 ; 1] |f(x) = x}
f
f f
u E u∗u
ker u∗= Im u⊥Im u∗= ker u⊥

E u ∈ L(E)u2= 0
ker(u+u∗) = ker u∩ker u∗
(e1, . . . , en)E
f
f(x) =
n
X
k=1
(ek|x)ek
∀x∈E\ {0E},(f(x)|x)>0
g E
g2=f−1
(g(e1), . . . , g(en)) E
x→1−
f(x) =
+∞
X
n=0
xn2
α∈]−1 ; 1[
x∈R
Pn(x) =
n
Y
k=0 1−αkx
P(x)
f:R→R
(E): ∀x∈R, f(x) = (1 −x)f(αx)
x∈R
f(x) = f(0)P(x)
x7→ P(x)R
p, q ∈N
I(p, q) = Z1
0
tp(1 −t)qdt
I(p, q)
un=I(n, n)
Punxn
n≥0
an=Zπ/4
0
tanntdt
(an)
an+2 an
f(x)
+∞
X
n=0
anxn
f
f
f(x) = Z+∞
0
e−t2cos(xt) dt
f