corrigé pdf

gn
[0,+∞[ [1,+∞[a > 1
αn>0gn(αn) = a
gn+1 gn
gn+1(αn) = gn(αn)+(n+ 1)αn+1
n=a+ (n+ 1)αn+1 n> a
gn+1 αn+1 < αn
gn(1) = 1 + 2 + ··· +n(gn(1))n∈N∗+∞
N gN(1) > a gN
αN<1
∀n≥N:αn≤αN<1
(αn)n∈N∗α
αn≤αN<1n≥N
0≤α≤αN<1
(npqn)n∈N∗
0p|q|<1
αn
αN
gn(x) = f0
n(x)fn(x) = 1−xn+1
1−x
fn
0 (xn)n∈N∗(nxn)n∈N∗
(gn(x))n∈N∗→1
(1 −x)2
(gn(x))n∈N∗n x
gn(x)≤1
(1 −x)2
n α ≤αngn
gn(α)≤gn(αn) = a
1
(1 −α)2≤a
β= 1 −1
√a
n
gn(β)≤1
(1 −β)2=a
gnβ≤αnβ(αn)n∈N∗
β≤α x →1
(1−x)2α≤β
α=β= 1 −1
√a
x=αna= 4 1 −αn=1
2(1 −εn)
4 = 1
(1 −αn)2−(n+ 1) αn
n
1−αn−α2
n
(1 −αn)n+1
⇒4(1 −αn)2= 1 −(n+ 1)αn
n(1 −αn)−αn+1 n
⇒(1 −εn)2= 1 −(n+ 1)1−εn
2αn
n−αn+1 n
⇒ −2εn+ε2
n=−(n+ 1)1−εn
2αn
n−αn+1 n
αn→1
2εn→0
ε2
nεn
αn+1
n(n+ 1)αn n

−2εn∼ −1−εn
2(n+ 1)αn
n
−1−εn
2(n+ 1)αn
n∼ −1
2nαn
n
εn∼1
4nαn
n
nεn∼1
4n2αn
n→0
(1 + εn)n→1
(1 + εn)n=enln(1+εn)nln(1 + εn)∼nεn→0
εn∼n
4
1
2n(1 + εn)n∼n 2n+2
q X y ∈X
fyX
∀x∈X:fy(x) = (1x=y
0x6=y
X={y1, y2,··· , yq}
f∈ F(X, C)
f=f(y1)fy1+f(y2)fy2+··· +f(yq)fyq
fy1, fy2,··· , fyq
F(X, C)
dim (F(X, C)) = q
p≤q
p V F(X, C)
dim (F(A, C)) = p= dim V
R
R
f∈ker R X C
A
0X A
x∈X−A x∗V∗
(x∗
1,··· , x∗
p)V∗λ1,··· , λpC
x∗=λ1x∗
1+··· +λpx∗ p
f∈V
x∗(f) = λ1x∗
1(f) + ··· +λpx∗ p(f)
f(x) = λ1f(x1) + ··· +λpf(xp) = 0
f A
f1,··· , fpA
∀(i, j)∈ {1,··· ,}2:fi(xj) = (1i=j
0i6=j
(f1,··· , fp)F(A, C)R
V
(v1,··· , vp)

∀i∈ {1,··· , p}:R(vi) = fi
∀(i, j)∈ {1,2,···}2:vi(xj) = (1i=j
0i6=j
V p
v∈V
x∈X
f(x)6= 0 ⇔x∗(f)6= 0
x∗V∗(x∗)
(x1,··· , xq)
(x∗
1,··· , x∗ q)
∀x∈X: (x∗
1,··· , x∗ q, x∗)
(x∗
1,··· , x∗
q)V∗p
V q ≤p
x∈X
x∗∈Vect x∗
1,··· , x∗ q
∀x∈X, ∃(λ1(x),··· , λq(x)) ∈Cqx∗=λ1(x)x∗
1+··· +λq(x)x∗ q
(λ1,··· , λq)XC
v∈V
∀x∈X, ∀v∈V:x∗(v) = λ1(x)x∗
1(v) + ··· +λq(x)x∗ q(v)
∀x∈X, ∀v∈V:v(x) = λ1(x)v(x1) + ··· +λq(x)v(xq)
∀v∈V:v=v(x1)λ1+··· +v(xq)λq
v∈Vect(λ1,··· , λq)
v∈Vdim V=p≤q p =q
(x∗
1,··· , x∗
p)V∗
PΦn(x1,··· , xn)
L= (X−x1)···(X−xn)
Φ
PΦ
L Q n −2
P=LQ
P0=LQ0+L0Q
∀i∈ {1,··· , n}−{k}: 0 = f P0(xi) = e L(xi)
=0 f
Q0(xi) + e L0(xi)
6=0 e
Q(xi)
n L n Q
n−1
δij 1i=j0i6=j
Li(xj) = δij
(L1,··· , Ln)n= dim Rn−1[X]
λ1L1+··· +λnLn
xiX i
λi= 0
P(L1,··· , Ln)
(e P(x1),··· ,e P(xn))

xjΛiΛi(xj) = 0 i6=k
xixk
Λ0
i(xj) =0 j6=j6=k
Λ0
i(xi) =(xi−xk)Y
j∈{1,··· ,n}−{i,k}
(xi−xj)2j=i
Λ0
i(xk) =(xk−xi)Y
j∈{1,··· ,n}−{i,k}
(xk−xj)2j=k
2n−2 = dim E
l1L1+···lnLn+λ1Λ1+···λnΛn= 0
xili
L xii6=k λi
T
T=l1L1+···lnLn+λ1Λ1+···λnΛn
T xi
l1=··· =lk= 1 lk+1 =··· =ln= 0
S=L1+··· +Lk
T0xi
i6=k
λi=−S0(xi)
Λ0
i(xi)
T
2n−2Lin−1 Λi2n−2
T0n−1xii6=k
x1x2x2x3
xk−1xkT
k−1ξ1,··· , ξk−1
x1< ξ1< x2< ξ2< x2<··· < xk−1< ξk−1< xk
xk+1, . . . , xn0
n−k−1ξk+1,··· , ξn−1
xk+1 < ξk+1 < xk+2 < ξk+2 < xk+2 <··· < xn−1< ξn−1< xn
T02n−3 2n−3
T0T0
x1max &ξ1min x2max &ξ2min ··· xk−1max &ξk−1min (1)
x1min %ξ1max x2min %ξ2max ··· xk−1min %ξk−1max (2)
xk+1 max &ξk+1 min ··· xn−1max &ξn−1min xnmax (3)
xk+1 min %ξk+1 max ··· xn−1min %ξn−1max xnmin (4)
xk
T0
]ξk−1, xk+1[
T xk
T(xk)=1 T(xk+1) = 0
T ξk−1
xk+1
(2) (4)
T
]−∞, x1[ ]xn,+∞[T(x1)=1 T(xn)=0
∀x≤x1:T(x)≥1∀x≥xn:T(x)≥0

T
xiT0
T
+∞T