Indications pour les leçons

Iα > 0 (un)n∈N∗u1>0
∀n≥1un+1 =un+1
nαun
(un)n∈N∗α > 1
un+1 −un
Ipne(p1= 2, p2= 3, p3= 5 ···)
X
n∈N∗
1
pn
If∈CM(R+,R)
R+∞
0f(t)dt t > 0
X
n∈N∗
f(nt)
t→0+

X
n∈N∗
xn2∼√π
2√1−xx→1−
Iα > 0
X
n∈N∗
(−1)n−1
nα+ (−1)n
α > 1/2
P(un+1 −un)
I
(x, θ)∈]−1,1[×R
X
n≥0
xncos(nθ),X
n≥0
xnsin(nθ)
I
z∈C|z|<1
X
n∈N∗
zn
(1 −zn)(1 −zn+1)

I[0, z0]
Panzn
0
+∞
X
n=1
(−1)n
n=−log 2 ;
+∞
X
n=0
(−1)n
2n+ 1 =π 4
I
2π
+∞
X
n=0
1
(2n+ 1)2=π2 8
2π
[0, π]
+∞
X
n=0
1
(2n+ 1)4=π4 96
+∞
X
n=1
1
n4=π4 90
I
Pn∈Nun(un)u0= 1
∀n∈N,un+1
un
=n+a
n+bb−a > 1
+∞
X
n=0
un=b−1
b−a−1
Pun
∀n∈Nu2n=1
n+ 1 u2n+1 =−1 n+ 1

I
PwnPun∀n∈Nun= (−1)n/√n+ 1
∀n∈N|wn|=
n
X
k=0
ukun−k
≥1.
I
Pn∈N∗
(−1)n−1
nα0< α ≤1
α > 0
X
n∈N∗
(−1)n−1
nα+ (−1)n
α > 1 1 ≥α > 1/2
1/2≥α > 0
[0, z0]
Panzn
0
+∞
X
n=1
(−1)n
n=−log 2 ;
+∞
X
n=0
(−1)n
2n+ 1 =π 4
I
X
n∈N∗
einθ
nα, α ∈]0,1] , θ ∈R\(2πZ)