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IR
(fn)f
g
(g◦fn)
(fn) (gn)
f g
(fngn)fg
(fn) [a;b]
fnf
inf
[a;b]fn→inf [a;b]f
(fn) [a;b]R
f: [a;b]→R(xn) [a;b]
x
fn(xn)−−−−−→
n→+∞f(x)
(Pn)R R
fRf
n∈N∗
un(x) = xnln x x ∈]0 ; 1] un(0) = 0
(un)n≥1[0 ; 1]
fn: [0 ; +∞[→R
fn(x) = x
n(1 + xn)
un(x) = e−nx sin(nx)x∈R+
(un) [0 ; +∞[
[a; +∞[a > 0
[0 ; +∞[
fn(x) = nx2e−nx x∈R+
(fn)R+[a; +∞[a > 0
fn(x) = 1
(1 + x2)nx∈R
(fn)R]−∞;−a]∪[a; +∞[
a > 0

fn(x) = x2sin 1
nxx∈R∗fn(0) = 0
(fn)R
(fn) [−a;a]a > 0
R(fn)
fn(x) = sinn(x) cos(x)
(fn)
fn(x) = nx2e−nx
1−e−x2
x≥0
fp(x) = 1
(1 + x)1+1/p
(fp)p∈N∗
fn(x) = 2nx
1 + n2nx2x∈R
fn(x)=4n(x2n−x2n+1 )x∈[0 ; 1]
α∈Rfn: [0 ; 1] →R
fn(x) = nαx(1 −x)n
(fn)
α∈R
n∈NfnR+
fn(x) = 1−x
nnx∈[0 ; n[fn(x)=0 x≥n
(fn)
fn:R+→R
fn(x) = 1 + x
n−n
(fn)
∀x∈R+, fn(x)≥lim fn(x)
t∈R+
t−t2
2≤ln(1 + t)≤t
(fn) [0 ; a]
a > 0
(fn)R+
fn: [0 ; 1] →R
fn(x) = n2x(1 −nx)x∈[0 ; 1/n]fn(x)=0
(fn)

Z1
0
fn(t) dt
(fn)
[a; 1] a > 0
x∈[0 ; π/2] fn(x) = nsin xcosnx
(fn)
In=Zπ/2
0
fn(x) dx
(fn)
]0 ; π/2]
fn(x) = x(1 + nαe−nx)R+
α∈Rn∈N∗f
α
lim
n→+∞Z1
0
x(1 + √ne−nx) dx
(fn)R+
f0(x) = x fn+1(x) = x
2 + fn(x)n∈N
(fn)n≥0R+
E f : [0 ; 1] →R+
Φ(f)(x) = Zx
0pf(t) dt
f∈E
f0= 1 fn+1 = Φ(fn)n∈N
(fn)
f= lim(fn)
f
fn:R+→R
fn(x) = x+ 1/n
(fn) (f2
n)
fn:R→R
fn(x) = px2+ 1/n
fnC1(fn)
RfC1
f:R→R
gn:x7→ n(f(x+ 1/n)−f(x))
f0
fn: [0 ; 1] →R(fn)

U ω
6= 1
z7→ 1
z−ω
U
f:R→RC1n∈N∗
un(t) = n(f(t+ 1/n)−f(t))
(un)n≥1
R
(un) [0 ; 1] R
u0(x) = 1 ∀n∈N, un+1(x) = 1 + Zx
0
un(t−t2) dt
x∈[0 ; 1]
0≤un+1(x)−un(x)≤xn+1
(n+ 1)!
x∈[0 ; 1] (un(x))
(un)u
u0(x) = u(x−x2)
x > 0
S(x) =
+∞
X
n=0
(−1)n
n+x
SC1R∗
+
S
∀x > 0, S(x+ 1) + S(x) = 1/x
S
S+∞
x > 0
F(x) =
+∞
X
n=0
(−1)n
n+x
F
FC1C∞
F(x) + F(x+ 1)
x > 0
F(x) = Z1
0
tx−1
1 + tdt
F+∞
x > 0
S(x) =
+∞
X
n=0
n
Y
k=0
1
(x+k)
S]0 ; +∞[
S(x)S(x+ 1)
S(x) +∞
n∈Nx∈R+
fn(x) = th(x+n)−th n

Pfn
S=P+∞
n=0 fn
R+
∀x∈R+, S(x+ 1) −S(x) = 1 −th x
S+∞
f:R+→R
g:R+→R
∀x∈R+, g(x+ 1) −g(x) = f(x)
∀x∈R,
+∞
X
n=0
xn
n!= ex
x > 0
S(x) =
+∞
X
n=0
(−1)n
n!(x+n)
SC1R∗
+
S
xS(x)−S(x+ 1) = 1
e
S+∞
S
f(x) = 1
x+
+∞
X
n=1
1
x+n+1 x−n
x∈R\Z
f
fx
2+fx+ 1 2= 2f(x)
x∈R\Z
f∈ C ([0 ; 1],R)
∀x∈[0 ; 1], f(x) =
+∞
X
n=1
f(xn)
2n
(E): f(2x)=2f(x)−2f(x)2
R
h:R→Rf(x) = xh(x)x∈R
h f (E)
R R
h0:x7→ 1n∈N
hn+1(x) = hnx
2−x 2hnx 22
x∈[0 ; 1] Tx:y7→ y−xy2/2Tx
[0 ; 1] Tx[0 ; 1]⊂[0 ; 1]
(hn)n∈N[0 ; 1]
(E)
[0 ; 1]
(E)
R+
n∈N∗unR∗
+
un(x) = xln1 + 1
n−ln1 + x n