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fR R
℘(N)N
ϕ
N℘(N)
A=n∈Nn /∈ϕ(n)
x
anxn+· · · +a1x+a0= 0 a0, a1, . . . , an∈Zan6= 0
x n ∈Nx
n
+∞
X
n=0
1
2n+1
(un)n∈N[0 ; 1]
∀n∈N,[0 ; 1] \
n
[
k=0
[uk−1
2k+2 ;uk+1 2k+2 ]6=∅
(xn)n∈N[0 ; 1]
∀n∈N, xn/∈
n
[
k=0
[uk−1
2k+2 ;uk+1 2k+2 ]
(xn)n∈N
`[0 ; 1]
[0 ; 1]
N
α > 0
1
(p2+q2)α(p,q)∈N∗2
z|z|<1
+∞
X
n=0
z2n
1−z2n+1
`1(Z)u= (un)n∈Z
u, v ∈`1(Z)n∈Z(ukvn−k)k∈Z
u, v ∈`1(Z) (u∗v)n=Pk∈Zukvn−ku∗v∈`1(Z)
X
n∈Z
(u∗v)n=X
n∈Z
unX
n∈Z
vn

∗
(`1(Z),∗)
q∈C|q|<1
q|n|n∈Z
r∈[0 ; 1[ θ∈R
X
n∈Z
r|n|einθ
Pn≥0unvn=uσ(n)σ∈ SN
Pn≥0vn
Pun
σ:N∗→N∗
X
n≥1
1
σ(n)2
X
n≥1
1
σ(n)
σN∗N∗
Xσ(n)
n2
σN∗
Xσ(n)
n2ln n?
(un)Pu2
n
σN(vn)
vn=uσ(n)n∈N
Pv2
n
P|unvn|
+∞
X
n=0
|unvn|
σN
(zn)
n6=m=⇒ |zn−zm| ≥ 1
1/z3
n
α > 1
Rn=
+∞
X
k=n+1
1
kα
α∈RP+∞
n=0 P+∞ k=n+1 1
kα

+∞
X
n=0
+∞
X
k=n+1
1
kα=
+∞
X
p=1
1
pα−1
a
up,q =ap(2q−1) p, q ≥1
+∞
X
p=1
ap
1−a2p=
+∞
X
p=1
a2p−1
1−a2p−1
ap,q =2p+ 1
p+q+ 2 −p p+q+ 1 −p+ 1 p+q+ 3
+∞
X
q=0
+∞
X
p=0
ap,q
+∞
X
p=0
+∞
X
q=0
ap,q
X
(p,q)∈N×N∗
1
(p+q2)(p+q2+ 1)
+∞
X
n=1,n6=p
1
n2−p2=3 4p2
+∞
X
p=1
+∞
X
n=1,n6=p
1
n2−p26=
+∞
X
n=1
+∞
X
p=1,p6=n
1
n2−p2
+∞
X
n=0
(n+ 1)3−n
(un)n∈N
vn=1
2n
n
X
k=0
2kuk
Pun
Pvn
Pun
(un)
(vn)
Pun
Pvn
Pun
e
+∞
X
n=1
(−1)n−1
n.n!=
+∞
X
n=1
Hn
n!
Hn=
n
X
k=1
1
k
(un)n∈Nn∈N
vn=1
2n
n
X
k=0
2kuk
(vn)n∈N
(un)n∈N

α∈Rn∈N
un=
n−1
X
k=1
1
kα(n−k)α
α un

ANϕ
n∈NA=ϕ(n)n
A
n∈A n /∈ϕ(n)A=ϕ(n)
n /∈A n /∈ϕ(n)n∈A
n
anxn+· · · +a1x+a0= 0 a0, a1, . . . , an∈Zan6= 0
Z∗×Zn
n
+∞
X
n=0
1
2n+1 =1 2
1
1−1/2
n
X
k=0
1
2k+1 <
+∞
X
k=0
1
2k+1 = 1
[0 ; 1]
(xn)
[0 ; 1]
[0 ; 1] (un)n∈N
(xn)n∈N`
[0 ; 1] N∈NuN=`
(xn)n∈N`
[`−1/2N+2 ;`+ 1/2N+2]=[uN−1/2N+2 ;uN+ 1/2N+2]
n≥N xn
ENEn
J0 ; nKN
E=[
n∈N
En
EnE
i1, . . . , ikAN
n(A)=2i1+· · · + 2ik.
2
1
(p+q)2≤1 p2+q2≤2 (p+q)2
1
(p+q)2α(p,q)∈N∗2