EXERCICES de mathématiques - Lycée Alfred Kastler de Cergy

n
P
k=p
qk=qp1−qn−p+1
1−qq6= 1
(a+b)3=a3+ 3a2b+ 3ab2+b3
a3−b3= (a−b)(a2+ab +b2)
a

n1
fnx fn(x) = x5+nx −1
fn
n1unfn(un)=0
fn+1(x)−fn(x)x(un)
un≤1
n
(un)
lim
n→+∞nunun
1
nn+∞
1
n−unn+∞
n x
fn(x) = xn+1
2(1 −x)n+1 2In=
1
R
0
fn(x)dx.
I0=Π
8
g[0,1] g(x) = (1 −x)
1
2[0,1]
G(x) = −2
3(1 −x)
3
2
n
In=2
3(n+1 2)(In−1−In).
n > 0 In=2n+ 1
2n+ 4In−1
nIn=3×5×7... ×(2n+ 1)
6×8×10 ×... ×(2n+ 4) ×Π 8.
(In)
nIn61
n+ 1.
(In)

n1vn=
n
P
k=1
1
k
∀k∈N∗,1
k+ 1 6
k+1
R
k
dt
t
∀n∈N∗, vn6ln(n)+1
(un)u0= 1
un+1 =un+1
un
(un)
k u2
k+1 −u2 ku2 k
∀n∈N∗, u2
n= 2n+1+
n−1
P
k=0
1
u2
k
∀n∈N∗, u2
n>2n(un)
n
un62n+2+1
2vn−1
n
u2
n62n+5
2+ln(n−1) 2
un∼√2n n →+∞
n∈NfnRfn(x) = xn
1 + x2
In=Z1
0
fn(x)dx
(In)
∀n∈N0≤In≤1
n+ 1
lim
n→+∞In
In+ In+2
∀n∈NIn+2 =1
n+ 1 −In
I2p+1 p∈N
I1I3I5
p∀p∈N∗,I2p+1 =
p
P
k=1
(−1)p−k
2k+ (−1)pI1
n∈N∗Sn=Z1
0
x−(−1)nx2n+1
1 + x2x
x−(−1)nx2n+1
1 + x2x
n x
Sn=
n−1
P
k=0
(−1)k
2k+ 2

n
Sn
Sn= (−1)n−1I2n+1 + I1
lim
n→+∞Sn
n≥1,Sn=
n
X
k=0
1
k!
e
e.
n≥1, n!≥n n!→
n→+∞+∞.
Tn≥1,Tn=
n
X
k=0
1
k!+1 n!= Sn+1 n!.
S T
n≥1 : Sn≤TnS T
S T ` n : Sn≤`≤Tn
n, Sn` ε
`SnS, k
K,1/k! F epsilon
n= 0 F,S
1/(k+ 1)! 1/k!
F K
k+ 1
S F
Sn
1/n!
`=e
n:
fn(x) = ex−
n
X
k=0
xk
k!gn(x) = ex−
n
X
k=0
xk
k!−(e−1) xn n!=fn(x)−(e−1) xn n!
n, f0
n+1 (x) = fn(x)n
n x ∈[0,1], fn(x)≥0
n x ∈[0,1], gn(x)≤0
fn(x)≤(e−1) xn
n!
x n :
0≤e−Sn≤e−1
n!
Snn+∞.

E = M3,1(R) F = M2,1(R)
f∀X = 
x
y
z
∈Ef(X) = 2x+ 3y−z
4x+ 3y−2z
f
f
f= F
(e1, e2, e3)
g g(e1) = e1+e2−2e3g(e2) = −e1−e2+e3
g(e3) = 2e1+ 2e2−2e3
g
g g
(xn) (yn) (zn) (tn)x0= 1 y0=1
2z0= 0 t0= 0
∀n∈N∗
xn=yn−1+zn−1+tn−1
yn=xn−1+zn−1+tn−1
zn=xn−1+yn−1+tn−1
tn=xn−1+yn−1+zn−1
Xn=
xn
yn
zn
tn
M∀n∈N∗Xn= MXn−1
M M−1
(a, b) M2=aM + bI
(an) (bn)∀n∈NMn=anM + bnI
an+1 bn+1 anbn
anbnn
Mnn
xnynzntn
N
5
0,1
N
X
Y
n
kP[N=n](X = k)
(X,N) n, k P([N = n]∩[X = k])
X 0,5
Y
i j P([X = i]∩[Y = j])
X Y