Familles sommables de nombres complexes - Chapitre de cours

N
Z
f:
N→Z
n→(n/2n
−(n+ 1)/2n
f
N
ENE f(0) E\
{f(0)}f(1) E\{f(0), f(1), ..., f(n−1)}
f(n)f:N→E
f(n+ 1) >f(n)+1 f
f q ∈E
fNlim
n→+∞f(n)=+∞ ∃p∈N, f(p)6q <
f(p+ 1) f(p+ 1) E\ {f(0, ..., f(p)}q∈E q =f(p)f
NE E
N
N2

n= 2 E F f :E→N
g:F→N(f, g) : E×F→N2N2
ϕ:N2→Nϕ◦(f, g)
E×FNE×F
E1×E2×... ×Ep+1 = (E1×... ×Ep)×Ep+1
Z2Zpp∈N∗
I∀i∈I, EiS
i∈I
Ei
Q
Q=S
q∈N∗p
q/p ∈ZEq=p q/p ∈ZEq
ZEqQ
PN
PnJ0, n −1KPn
P Pn
R
I
(ui)i∈II
(X
i∈F
ui/ F I )
R+

X
i∈I
ui
X
i∈I
ui= +∞
1
q2q∈Q∩[1,+∞]
E=Q∩[1,2] E
n∈NFnEX
q∈Fn
1
q2>n 4
(ui)i∈I
J⊂I(ui)i∈JX
i∈J
ui6X
i∈I
ui
(X
i∈F
ui/ F J )⊂(X
i∈F
ui/ F I ) (X
i∈F
ui/ F J )
(X
i∈F
ui/ F I ) (X
i∈F
ui/ F J )
(ui)i∈I(vi)i∈I∀i, ui6vi
•(vi)i∈I(ui)i∈I06X
i∈I
ui6X
i∈I
vi
•(ui)i∈I(vi)i∈I
(In)n∈NI
(ui)i∈I
(ui)i∈I⇐⇒ 
∀n∈N,(ui)i∈In
X
n X
i∈In
ui!
X
i∈I
ui=
+∞
X
n=0 X
i∈In
ui!
1
(m+n)3(m,n)∈N∗2
p∈N∗Ip=(m, n)∈N∗2/m +n=p
(Ip)p∈N∗N∗2

(ui)i∈I
(vi)i∈I=⇒(ui+vi)i∈I
X
i∈I
(ui+vi) = X
i∈I
ui+X
ß∈I
vi
(ui)i∈I
α>0=⇒(αui)i∈I
X
i∈I
(αui) = αX
i∈I
ui
I
(ui)i∈I
(|ui|)i∈I
•(ui)i∈I
I+={i∈I/ ui>0}I−={i∈I/ ui<0}
X
i∈I
ui=X
i∈I+
|ui| − X
i∈I−
|ui|
•(ui)i∈I
X
k∈I
uk=X
k∈I
Re(uk) + iX
k∈I
Im(uk)
(ui)i∈I(|ui|)i∈I
(|ui|)i∈I+(|ui|)i∈I−
(ui)i∈I(|uk|)k∈I
|Re(uk)|6|uk| |Im(uk)|6|uk|(Re(uk))k∈I(Im(uk))k∈I
(ui)i∈IJ⊂I, (ui)i∈J
J
(uk)k∈I(vk)k∈I
α β (αuk+βvk)k∈I
X
k∈I
(αuk+βvk) = αX
k∈I
uk+βX
k∈I
vk
I
(uk)k∈I→X
k∈I
uk

I=N(un)n∈N
(un)n∈N⇐⇒ X
n
un
X
n∈N
un=
+∞
X
n=0
un
I=N(un)n∈N
(un)n∈N⇐⇒ X
n
un
X
n∈N
un=
+∞
X
n=0
un
Xunσ
N N X
n
uσ(n)
+∞
X
n=0
uσ(n)=
+∞
X
n=0
un
(In)n∈NI
(ui)i∈I
(ui)i∈I⇐⇒ 
∀n∈N,(ui)i∈In
X
n X
i∈In
|ui|!
X
i∈I
ui=
+∞
X
n=0 X
i∈In
ui!
(ui)i∈I⇐⇒ (|ui|)i∈I
(|ui|)i∈I⇐⇒ 
∀n∈N,(|ui|)i∈In
X
n X
i∈In
|ui|!⇐⇒ 
∀n∈N,(ui)i∈In
X
n X
i∈In
|ui|!
X
nX
i∈In
ui