Séries entières - IMJ-PRG

+∞
X
n=0
(n+ 1) xn
+∞
X
n=0
√n xn
+∞
X
n=0 n+1
n+ 1xn
+∞
X
n=0
nnxn
+∞
X
n=0
1
√n√nxn
+∞
X
n=0
ln(n+ 1) xn
+∞
X
n=0
2n
n2+ 1 xn
+∞
X
n=0
1
(3n)! x2n+1
+∞
X
n=0
sin nπ
2013xn
k(n+ 1)k=nk+nkε(n)ε
0 +∞
P=
d
X
k=0
pkxkP(n+ 1) ∼
n→+∞P(n)
+∞
X
n=0
P(n)xn

+∞
X
n=0
x3n
+∞
X
n=0
(−1)n
(2n)! xn
+∞
X
n=0
n2xn
+∞
X
n=1
x2n
2n
+∞
X
n=0
((−2)n+n)xn
+∞
X
n=0
cos(nθ)
n!xn
+∞
X
n=1
xn
n(n+ 2).
f
f0f
1
n(n+2) f
+∞
X
n=0
(−1)nxn
n+ 1 .
R−R R
]−R, R[
f(x) = arctan(1 −x)
f
f0(x) = i
21 x−(1 + i)−1 x−(1 −i)
x∈]−√2,√2[
f0(x) = i
√2
+∞
X
n=0
cos (n+ 1)π
41
√2n
xn.
f0
xy0+ 2y=x3+ 1.
y0−2xy =x
y(0) = 0

y00 + 2xy0+ 2y= 0
y(0) = 1 y0(0) = 0
(un)n∈Nu0=u1= 1
un+1 =un+un−1
unn S(x) =
+∞
X
n=0
unxn
(un)n∈N
n≥1un+1
un≤2
S(x) = 1 + (x2+x)S(x)S(x) = 1
1−X−X2
1
1−X−X2un
n
f k ∈N∗
f(1
k) = 0 f
"
IRa
I f :I→Rn f n
I ε(x)I0
x a x ∈I
f(x) = f(a)+(x−a)f0(a) + (x−a)2
2! f00(a) + ··· +(x−a)n n!f(n)(a)+(x−a)nε(x).
"