Exercices de révision MP* : Maths, Analyse, Algèbre

C
1 + eit 1−eit
C
n1zn=a
c−a
b−a

(xn)n≥0
∀n∈N, xn+1 =axn+b.
xnx0, a, b, n
x∈R
C(x) =
n
X
k=0 n
kcos(kx), S(x) =
n
X
k=0
sin(kx).
nN∗
PCn−1[X]
P(0) = 1
nX
ω∈Un
P(ω).
|P(0)| ≤ max{|P(ω)|;ω∈Un}
nN∗ω= exp(2iπ
n)Q
Q[X]n−1Q(ω)=0.
x= 2 cos(2π/5)
x=ω+ 1/ω, ω = exp 2iπ
5,
2x
x
j=e2iπ/3a, b, c
a+bj +cj2= 0.
c b

1∞0×∞
ean∼ebn⇔an−bn→0.
∀x > 0,(x)+ (1/x) = π/2
(1 −x)∼√2x x →0+
α
e(n+1)α∼enα?
4 0 ln(cos(x))
2x
ex−1·
2 exp(ex)·
1
tan x+1 sin x−2 x·
(cos x)
2x.
+∞x√x
√xx.
+∞2
πxx
+∞e1/x −x(x+ 1)
1 + x2.
π/4 (tan x)tan 2x.
a b a > 0 +∞
ln (ln(ax +b)) −ln (ln x).
αR+∗+∞(x)α−(x)α.
u v R+∗(an) (bn)R+∗
ann→u, bnn→v.
an+bn
2n
.

f:R→R
x7→ x5+x
fR R f−1
C∞
f−1(x)x→+∞
n∈N∗xn+x= 1
xnR+
(xn)`
xn−`
tan x=x n ∈N
xn]−π/2 + nπ, π/2 + nπ[
xn
±∞
un+1 =f(un)
R C
zC(zn)n≥0
(xn)n≥0nN
yn=1
n+ 1
n
X
k=0
xk.
(xn)n≥0`
(yn)n≥0(xn)n≥0+∞
(xn)n≥0(yn)n≥0`
(xn)n≥0`
d≥2
∀n∈N, xn+d=xn.
(yn)n≥0

f C1IR
a I f
|f0(a)|<1,
(un)n≥0I
∀n∈N, un+1 =f(un).
k]|f0a)|,1[ α > 0f k
I∩[a−α, a +α]
u0I∩[a−α, a +α] (un)n≥0a
u0
a(un)n≥0
R
(xn)
ϕ
xϕ(n)→+∞.
(un)n≥0
un+1 −un→0.
(un)≥0
a b a < b c ]a, b[
c
f1
nf C2[0,1]
f(n)f
exp(−(ln n)α)α > 0,