Convergence des suites numériques. Exemples et

R
R
Qε∈Q+N
n, m >N→ |un−um|6ε
EQE
0
R
Q R r
r
R+∩ −R+={0}
x6y⇔y−x∈R+R
R C
un→l⇔ε > 0∃N > 0n6N⇒ |un−l|6ε
ε > 0n6N⇒ |un−l|6ε
an+am6an+man/n
1/nnPkne/(e−1)
un=pa0+√a1+. . . una1/2n
nlimp1+2√1+3. . . = 3

¯
R
(−1)n
uk=Pn
k=1(−1)kCk
nln(k)
unun
limn60supk6nun+∞
T¯
XnXn=∪k>nuk
ε|un−l|6n
z > limsup
un< z z < limsup un> z
un
un+u2n/2en
an+bn↔0ean+ebn↔2
0an+bn+cn↔0ean+ebn+ecn↔3
0
xnt eitxn
unun
unun
un⇒unsiAn
+∞An
l
un0A⊂N
limn6∈Aan= 0
p1Ep
unl
p
un+1 =f(un)vn
f

(un+1 −a)/(un−a)
an−l=vn+o(vn)an−vn
l an
anPn
k=1 1/k2n1/n
an=π2/6−1/n an=π2/6−1/n +O(1/n2)
un=
λ+akn+o(hn)vn=un+1 −kun/(1 −k)O(h)
un+1 =f(un)
f C20x0
un+1 =F(un)F(x) = x−f(x)/f0(x)
0f00(x)>0
f(x) = x2−y. F (x)−a= (x−a)2/2x
2n
A
λ
v0=v
vn+1 =Avn//||Avn||
vn(vn+1, vn)

A λ µ
uk+1 = (A−µI)−1uk/||uk|| uk
0ci
P(ai+1 −ai)f(ci)Rb
af(t)dt