Intégrabilité et comportement asymptotique

f: [0 ; +∞[→RC1
f2f02f+∞
f: [0 ; +∞[→R
f
Zx+1
x
f(t) dt−−−−−→
x→+∞0
f:R+→R R+
f+∞
xf(x)x→+∞
f
R+f+∞
f: [0 ; +∞[→R
(xn)
xn→+∞xnf(xn)→0
f∈ C0(R+,R+)
f∈ C2([0 ; +∞[,R)f f00
f0(x)→0x→+∞
f.f0
g:R+→R
∀ε > 0,∃M∈R,
Z+∞
0
g(t) dt−ZM
0
g(t) dt
≤ε
g
fC2[0 ; +∞[f00 [0 ; +∞[
R+∞
0f(t) dt
lim
x→+∞f0(x) = 0 lim x→+∞f(x)=0
Xf(n)Xf0(n)

ab ≤1
2(a2+b2)
|ff0| ≤ 1
2f2+f02
ff0[0 ; +∞[
Zx
0
ff0(t) dt=1
2(f(x))2
f2x→+∞f2
[0 ; +∞[ +∞f−−→
+∞0
Zx+1
x
f(t) dt=Zx+1
0
f(t) dt−Zx
0
f(t) dt
x→+∞
Zx+1
x
f(t) dt→Z+∞
0
f(t) dt−Z+∞
0
f(t) dt= 0
x≥1f
Zx+1
x
f(t) dt≤f(x)≤Zx
x−1
f(t) dt
Zx+1
x
f(t) dt=Zx+1
0
f(t) dt−Zx
0
f(t) dt
f[0 ; +∞[
Zx+1
x
f(t) dt−−−−−→
x→+∞Z+∞
0
f(t) dt−Z+∞
0
f(t) dt= 0
Zx
x−1
f(t) dt−−−−−→
x→+∞0
f(x)−−−−−→
x→+∞0
f+∞
0≤x
2f(x)≤Zx
x/2
f(t) dt−−−−−→
x→+∞0
xf(x)−−−−−→
x→+∞0
fR+
∀x∈[0 ; 2[, f(x)=0
∀n∈N\ {0,1},∀t∈[0 ; 1[, f(t+n) = 
n2t t ∈[0 ; 1/n2]
n2(2/n2−t)t∈[1/n2; 2/n2]
0
fR+
Zn
0
f(t) dt=
n−1
X
k=0 Zk+1
k
f(t) dt=
n−1
X
k=2
1
k2≤
n−1
X
k=2
1
k(k−1) =
n−1
X
k=2
1
k−1−1 k= 1−1 n−1≤1
([0 ; n])n∈N
R+f f [0 ; +∞[
∀ε > 0,∀A∈R+,∃x≥A, |xf(x)| ≤ ε
ε > 0A∈R+
∀x≥A, |xf(x)| ≥ ε
+∞
|f(x)| ≥ ε
x

f
∀ε > 0,∀A∈R+,∃x≥A, |xf(x)| ≤ ε
(xn)ε= 1/(n+ 1) >0A=n
xn
xn≥n|xnf(xn)| ≤ 1/(n+ 1)
f[0 ; 1] [n;n+ 1]
n∈N∗f f(n)=0
f(n+1
n3) = n f(n+2 n3)=0 f(n+ 1) = 0 f
Rn+1
nf=1
n2R+
f0(x) = f0(0) + Zx
0
f00(t) dt
f0(x)` x →+∞
` > 0x f0(x)≥`/2f(x)≥`x/2 + m
R+∞
0f(t) dt
` < 0`= 0
f0+∞
t7→ f(t)f0(t) [0 ; +∞[
limM→+∞RM
0
g(t) dt=R∞
0
g(t) dt
f g f(x) = Rx
0g(t) dt+C
x≤y∈R|f(y)−f(x)| ≤ Ry
x
g(t) dt
ε > 0M
x≥M
f(y)−f(x) ≤Z+∞
M
g(t) dt≤ε
t7→ |g(t)|[0 ; M+ 1]
A
f(y)−f(x) ≤A|y−x|
x≤y∈[0 ; M+ 1]
α= min(1, ε/A)>0x≤y∈R
|y−x| ≤ α=⇒ |f(y)−f(x)| ≤ ε
f
fC2
f0(x) = f0(0) + Zx
0
f00(t) dt
f00 f0`
x→+∞
` > 0x f0(x)≥`/2A≥0
x≥A
f(x) = f(0) + Zx
0
f0(t) dt≥f(0) + ZA
0
f0(t) dt+Zx
A
`
2dt
f(x)≥`x/2 + Cte R+∞
0f(t) dt
` < 0`= 0
F(x) = Zx
0
f(t) dt
F(x+ 1) = F(x) + f(x) + Zx+1
x
(x+ 1 −t)f0(t) dt
x→+∞
F(x), F (x+ 1) →Z+∞
0
f(t) dt
f0(x)→0
Zx+1
x
(x+ 1 −t)f0(t) dt ≤max
t∈[x;x+1] |f0(t)| → 0
f(x)→0

f(n+ 1) = f(n) + f0(n) + Zn+1
n
((n+ 1) −t)f00(t) dt
f0(n) = f(n+ 1) −f(n) + Zn+1
n
(n+ 1 −t)f00(t) dt
f(n+ 1) −f(n)
(f(n)) +∞
Rn+1
n(n+ 1 −t)f00(t) dt
Zn+1
n
(n+ 1 −t)f00(t) dt ≤Zn+1
n
|f00(t)|dt
f00
Pf0(n)
F(n+ 1) = F(n) + f(n) + 1
2f0(n) + Zn+1
n
(n+ 1 −t)2
2f00(t) dt
Pf(n)