Exercice - Alain Camanes

2015/2016
Exercice 1. (-)a < b f ∈F(]a, b[,R)
f a =−∞
Exercice 2. (-)f∈F(R+,R)f
R
Exercice 3. (-)f∈F(R,R) lim
+∞f=`lim −∞ f=`0f
R
Exercice 4. ( ! )f∈C([0,1],R) lim
n→+∞
1
n
n
P
k=1
(−1)kfk
n
Exercice 5. x7→ Zx
0btcdt
Exercice 6. (-)a∈[1,+∞[f∈C1([1,+∞[,R)
Za
1btcf0(t)dt =bacf(a)−bac
X
k=1
f(k).
Exercice 7. (♥)p, q Ip,q =Z1
0
xp(1 −x)qdx
1. Ip,q Ip+1,q−1
2. Ip,q
Exercice 8. (♥)f∈C([a, b],R)f
Z[a,b]
f=Z[a,b]|f|
Exercice 9. (Point fixe) f∈C([0,1],R)
1. Z[0,1] f= 0 f]0,1[
2. Z[0,1] f=1
2f]0,1[
Exercice 10. f∈C([a, b],R)n∈Nk∈J0, nKZb
a
tkf(t)dt = 0
1. P∈Rn[X]Zb
a
P(t)f(t)dt = 0
2. f n + 1 [a, b]

Exercice 11. (πest irrationnel, ♥)a, b, n ∈N?Pn=1
n!Xn(bX −a)n
1. Pn0a
b
2. In=Zπ
0
Pn(x) sin(x)dx lim
nIn= 0
3. π a b In∈Z
4. π
Exercice 12. a∈R?
+f∈C1([0, a],R)f(0) = 0 f
g:x7→ Zx
0|f0(t)|dt
1.
1
2g(a)26a 2Za
0
g0(t)2dt.
2. x∈[0, a]|f(x)|6g(x)
3. Za
0|f0(t)f(t)|dt 6a
2Za
0|f0(t)|2dt
Exercice 13. (Un air de Cesaro, ♥)f∈C(R+,R)F:R?
+→R, x 7→ 1
xZx
0
f(t)dt
1. F0
2. f ` +∞F ` +∞
Exercice 14. f∈C([0,1],R)n un=Z1
0
f(tn)dt
1. u f(0)
2. f0v n
vn=n(un−f(0))
Exercice 15. a > 0f:R→Rx > 0
I(x) = 1
xa+1 Zx
0
taf(t)dt
1. lim
x→0I(x)
2. f0I0I0(0)
Exercice 16. ( ! )f∈C([a, b],R+) lim
n→∞ Zb
a
f(x)ndx1
n
= sup
[a,b]
f
Exercice 17. ( ! )f: [0,1] →RC1f(1) = f0(1) = 0
lim
nn2Z1
0
xnf(x)dx

Exercice 18. (Sommes de Riemann, -)
1. S(1)
n=1
n
n−1
P
k=0
cos2kπ
n
2. S(2)
n=1
n√n
n
P
k=1b√kc
3. S(3)
n=1
nα
n
P
k=1
k2e−k2
n2
4. S(4)
n=
n
Q
k=1 1 + k2
n21
n
Exercice 19. (Intégrale de Poisson) x∈]1,+∞[I(x) = Zπ
0
ln(1 −2xcos t+x2)dt
f
1. I
2. n∈N?π
n·n−1
P
k=0
fkπ
n
3. I
Exercice 20. g∈C([−1,1],R)f[−1,1] f(x) = g(x)x6= 0
f(0) = g(0) + 1 Zx
0
f(t)dt 0F0(0) 6=f(0)
Exercice 21. (Fonction de Thomae) [0,1] f f(x) = 1
qx∈Q∩[0,1]
p
qf(x)=0
1. Aε={x∈[0,1] ; f(x)>ε}
2. f[0,1]\Q0
3. f]0,1] ∩Q
4. ε > 0ϕεϕε(x) = ε x 6∈ Aεϕε(x) = f(x)ϕε
Z1
0
ϕε(t)dt =ε06f6ϕεf
Z1
0
f(t)dt = 0