Suites et séries de fonctions

ζ

(fn)n∈NA f x ∈A
fn(x)−→
n→+∞f(x)
fn:x∈R7→ x
n
n∈Nx>0fn(x) = xn
1 + xn
xn
1+xnn∈ {1,5,20,50}
(fn)n∈Nf:x7→ 
0 si x∈[0,1[
1
2si x= 1
1 si x∈]1,+∞[
x ε > 0n|fn(x)−f(x)|6ε n
x|fn(x)−f(x)|>1
4x1
∀x∀ε, ∃N > 0; ... N ε
x
n∈Nx > 0fn(x) = Arctann+x
x(fn)n∈N
π
2·
(fn)n∈Nfn(x) = −xn
P
k=0
xk
k!
R+1
Arctann+x
xn∈ {1,5,20,50}−xn
P
k=0
xk
k!n∈ {1,5,10,15}

(fn)n∈Nf fn
cv.s.
−→
n→+∞f
∀x∈A, ∀ε > 0,∃N∈N;n>N=⇒ |fn(x)−f(x)|6ε.
N ε x fn(x) = xn
1 + xn
ε > 0x1N
|fn(x)−f(x)|6ε N x
(fn)n∈Nf fn
cv.u.
−→
n→+∞f
ε > 0x∈A|fn(x)−f(x)|6ε
∀ε > 0,∃N∈N;∀x∈A, n >N=⇒ |fn(x)−f(x)|6ε.
+∞
x∈A
|fn(x)−f(x)| −→
n→+∞0.
f A kfk∞kfk∞,A
|f|A
fn
cv.u.
−→
n→+∞f fn−fkfn−fk∞−→ n→+∞0
n∈Nx∈Rfn(x) = xn
(fn)n∈N[0,1] [0,1/2] [0,1−ε] 0 <ε<1 [0,1[
fn(x) = x
nfn
cv.s.
−→
n→+∞f= 0 Rfn−f=fn
S= [−M, M]kfn−fk∞,S =S
n−→
n→+∞0
fn(x) = xn
1 + xn
kfn−fk∞=1
2fn−f1
[0,2] ε0>0S= [0,1−ε0]
kfn−fk∞,S =fn(1 −ε0)−→
n→+∞0,
[0,1−ε0]

xn
1+xn[0; 0,9] n∈ {1,5,20,30}
fn(x) = Arctann+x
xkfn−fk∞=π
4
I=]0, M]M > 0
kfn−fk∞,I =fn(M)−π
2−→
n→+∞0
]0, M]
R+fn(x) = −xn
P
k=0
xk
k!
f1kfn−fk∞= 1
[0, M]R+
P
n∈N
fnPfngn=
n
P
k=0
fk
PfnxPfn(x)
Rn=
+∞
P
k=n+1
fn0
Pfn(fn)n∈N
R|cos x+ sin x|
cos(x2) + sin x
Rn
x|fn(x)|αnPαn
kRnk∞6
+∞
P
k=n+1
αk−→
n→+∞0
P
n∈N
fnI
fnP
n∈N
kfnkn∈N

Rn
kRnk∞−→
n→+∞0
x∈A fn
|fn(x)|6kfnk∞P|fn(x)|Pfn(x)
x∈A n fnk>n|fk(x)|6kfkk∞
p>1
n+p
X
k=n+1
fk(x) 6
n+p
X
k=n+1
|fk(x)|6
n+p
X
k=n+1
kfkk∞6
+∞
X
k=n+1
kfkk∞
n+p
P
k=n+1
fk(x) 6
+∞
P
k=n+1
kfkk∞p
|Rn(x)|p+∞
|Rn(x)|6
+∞
P
k=n+1
kfkk∞x∈A Rn
kRnk∞6
+∞
P
k=n+1
kfkk∞
kRnk∞0
n+∞
x∈Rn∈Nfn(x) = xngn(x) = xn
n·
(fn)n∈N
]−1,1]
[−1 + ε, 1−ε] 0 <ε<1
]−1,1]
Pfn
]−1,1[
[−1 + ε, 1−ε]
0<ε<1
]−1,1[
(gn)n∈N[−1,1]
Pgn
[−1,1[
[−1,1−ε] 0 <ε<1
[−1,1[
[−1 + ε, 1−ε] 0 <ε<1
]−1,1[