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unA
n
un6= 0 |A−(u0+u1+· · · +un)|≤|un+1|
A(θn)n≥1
]0 ; 1[ n
A−(u0+u1+· · · +un) = θn+1un+1
A > 0
unA
A
un
n∈N∗
A−(u0+u1+· · · +un)un+1
A
unA
|un|
A
E(un)n≥0
un+2 = (n+ 1)un+1 +un
E
a b E
a0= 1
a1= 0 b0= 0 b1= 1
(an) (bn) +∞
wn=an+1bn−anbn+1
cn=an/bnn
`= lim
n→+∞cn
r
lim
n→+∞(an+rbn)=0
e= (en)n∈N
Pen
s=
+∞
X
n=0
enrn=
+∞
X
k=n+1
ekn∈N
G=(+∞
X
n=0
dnen|(dn)∈ {−1,1}N)
e G
G
(en)
en≤rnn∈N
en≤rnn∈Nt∈[−s;s]
(tn)
t0= 0 tn+1 =(tn+entn≤t
tn−en
|t−tn| ≤ en+rn

en= 1/2nn∈N
G(dn)
0,1,1/2,2 1/3
x∈G(dn)∈ {−1,1}N
x=
+∞
X
n=0
dnen

PunPvn
Pun+vn
Pun|un|≤|un+vn|+|vn|
Pun+vn
k(k−1)
2< n ≤k(k+ 1) 2
un=(−1)k−1
k
(un)n≥1
1,−1
2,−1 2,1 3,1 3,1 3,−1 4,−1 4,−1 4,−1 4, . . .
un= (−1)nun
[0 ; 1]
un= (−1)n/(n+ 1) un
ln 2
un= 1/2nunun→0
+∞
X
k=0
1
2k−
n
X
k=0
1
2k=
+∞
X
k=n+1
1
2k=1 2n
un+1
Sn=
n
X
k=0
uk
θn+2un+2 =A−Sn+1 =A−Sn−un+1 = (θn+1 −1)un+1
θn+2 >0θn+1 −1<0un+2 un+1
A−Sn=θn+1un+1 un+1 A−Sn
A−Sn+1 A SnSn+1
A−Snun+1 A−Sn=θn+1un+1
θn+1 ∈R+
A−Sn+1 = (θn+1 −1)un+1 un+2 un+1
un+2 θn+1 −1≤0θn+1 ∈[0 ; 1]
A−Sn
n∈N
un= (−1)nA= 1 un
n A −Sn= 1 −1 = 0 un+1
n A −Sn= 1 −0 = 1 un+1
A−Sn6= 0 n∈Nθn+1 ∈]0 ; 1[
un+1, un+2 >0
A−Sn≤un+1 A−Sn+1 ≤0A−Sn+2 ≤ −un+2
|A−Sn+2|≥|un+2| |A−Sn+2| ≤ |un+3| |un+3|<|un+2|
un+1 un+2
un
A−Sn+1 =A−Sn−un+1
− |un+1| − un+1 ≤A−Sn+1 ≤ |un+1| − un+1
un+1 >0A−Sn+1 ≤0un+2
un+1 <0A−Sn+1 ≥0un+2
A−Sn+1
A−Sn+3 =A−Sn+1 −(un+2 +un+3) = −(un+2 +un+3)
un+2 un+4
unA
ERN
ϕ:E→R2ϕ(u)=(u0, u1)
E
E

(an) (bn)
∀n≥2, an, bn≥1
an+2 ≥n+ 1 bn+2 ≥n+ 1
(an) (bn) +∞
wn+1 = ((n+ 1)an+1 +an)bn+1 −an+1 ((n+ 1)bn+1 +bn)
wn+1 =−wn
wn= (−1)nw0= (−1)n+1
cn+1 −cn=wn
bnbn+1
=(−1)n+1
bnbn+1
bnbn+1 +∞
P(cn+1 −cn) (cn)
`−cn=
+∞
X
k=n
(ck+1 −ck)
|`−cn| ≤ 1
bnbn+1
cn=`+ o 1
bnbn+1 
an+rbn=bn(cn+r) = bn(`+r)+o1
bn+1 
bn→+∞
an+rbn→0⇐⇒ r=−`
|dnen| ≤ enPen
G
G
+∞
X
n=0
dnen ≤
+∞
X
n=0
en=s
G⊂[−s;s]s∈G(dn)n∈N= (1)n∈N−s∈G
(dn)n∈N= (−1)n∈N
e G = [−s;s]
N∈NeN> rN
x=
N−1
X
k=0
ek∈[−s;s]
x= 0 N= 0
y=
+∞
X
n=0
dnendn∈ {−1,1}
k≤N dk=−1
y≤
+∞
X
n=0
dnen−2ek=s−2ek
ek≥eN
y < s −2eN=x+rN−eN< x
dk= 1 k≤N
y=
N
X
n=0
ek+
+∞
X
n=N+1
dkek≥x+eN−rN> x
y6=x x /∈G
n∈N
n= 0
|t−t0|=|t| ≤ s=e0+r0

n≥0
tn≤t
t−tn+1 =t−tn−en≤rn
t−tn+1 ≥ −en≥ −rn
|t−tn+1| ≤ rn=en+1 +rn+1
tn> t
tn+1 −t=tn−t−en
tn→t t ∈G
e∀n∈N, en≤rn
s= 2 G= [−2 ; 2]
0 = 1 +
+∞
X
n=1
(−1) 1
2n,1 = 1 + 1 2+
+∞
X
n=2
(−1) 1
2n,1 2= 1 −1 2−1 4+
+∞
X
n=3
1
2n
2 =
+∞
X
n=0
1
2n
+∞
X
n=0
(−1)n
2n=2 3
1
3= 1 −1 2+
+∞
X
n=2
(−1)n−1
2n
(dn)x=s
x= 0 (dn) (−dn)