1 Suites et nombres réels

uN R u(n)
n un(un)n∈Nu
(un)l∈R∀ε > 0∃N∈N
∀n>N|un−l|6ε l l
(un)
(n
√n)n∈N∗1
(un)vn=u1+···+un
n
(un)
(un)∀ε > 0∃N∈N∀p, q >N|up−uq|6ε
Q
un= 1 + 1
1! +··· +1 n!Q Q
R
R
(un)
un= 1 + 1
2+··· +1 n
u2n−un>1
2
R
(un)n un6un+1 n
un>un+1
(un)M > 0n un6M
un

(un) (vn)
vn−un→0
un=Pn
k=0 1
k!vn=un+1 nn!
(un)vn= supk>nuk(vn)
l−∞
(un) (un)
(un)R=R∪ {±∞}
((−1)n) 1 −1
(un) (uϕ(n))ϕ:N→N
(un)
(un)
R
α∈R+\Q(rn)α n rn=pn
qn
pn∈Nqn∈N∗pnlim pn= lim qn=∞
ERE
E E
f K Rf
K
f K Rf
K
p x ∈[0,1[ p>2
(xn)n>1{0, . . . , p −1}x=P∞
n=1
xn
pn
{n∈N∗/ xn6=p−1}P∞
n=1
xn
pnx
p
R
Q R
b10nxc
10nx
1
10n(un)
e1
nn!

(un)k∈N∗Rk
Rn>k un=f(un−1, . . . , un−k)
(un)un=f(un−1, . . . , un−k)l f
(l, . . . , l)l=f(l, . . . , l)
un+1 =un+r
un+1 =qun
un=a1un−1+··· +akun−k
a1, . . . , ak∈RFn+2 =Fn+1 +FnF0=F1= 1
(un)n
1IRf:I→R
(un)u0∈I un+1 =f(un)
f(un)u1−u0
f(x) = √1 + x(un) Φ = 1+√5
2
f f ◦f(u2n) (u2n+1)
f(x) = 1 + 1
x(un)
Φ
IRf:I→IC2
k∈]0,1[ ∀x∈I|f0(x)|< k f λ
xn+1 =f(xn)x0∈I λ K =f0(λ)>0
xn> λ n c > 0xn−λ∼cKn
IRa∈˚ I f :I→R
a f(a)=0 f0(a)6= 0 f00
xn+1 =xn−f(xn)
f0(xn)a x0a
f
I f0(a)>0x0>a
y > 0f:x7→ x2−y xn+1 =1
2xn+y xn
√y x0>√y x0>max(y, 1))
f:x7→ arctan(x)f(x)=0 x= 0
xn+1 =xn+ (1 + x2
n) arctan(xn) arctan(|x0|)>2|x0|
1+x2
0(|xn|)
(xn)
u v

u v ∃M > 0∀n|un|6M|vn|u= (v)
u v ∀ε > 0∃N∀n>N|un|6ε|vn|u= (v)
u v u −v= (v)
un+1 = sin(un) 0
un∼q3
n
n!∼n
en√2πn
(un)Pun
(Pn
k=0 uk)n
(un)P(un+1 −un)
(un)>0un+1
un→k
k < 1Punk > 1
(un)>0L= lim sup n
√un
L < 1PunL > 1
Pxn
n!Pnln(n) ln(n)n
⇒(un)>0un+1
un
ln
√un→l
un= 2(−1)n−n
[a, b]R
f: [a, b]→RC1Rb
af(t) dt= limn→∞ b−a
nPn−1
k=0 fa+kb−a
n
1
n
lim P2n
k=n+1 1
k= ln(2) lim n
q(n+1)...(n+n)
nn=4 e
Rb
af(t) dt= lim b−a
nPn−1
k=0 fa+k+1
2b−a n  1 n2
1
n2
1
n4