Développements en séries entières usuels

Formulaire PanaMaths (CPGE)
Développements en séries entières usuels
Fonction
Développement en série entière
(DSE)
Intervalle de
validité du DSE
+∞
xn
x 2 x3
= 1 + x + + + ...
∑
2 6
n=0 n !
\
x 6 chx
x2n
x2 x4
x6
=
1
+
+
+
+ ...
∑
2 24 720
n = 0 ( 2n ) !
\
x 6 shx
x 2 n +1
x3 x5
x7
x
=
+
+
+
+ ...
∑
6 120 5040
n = 0 ( 2 n + 1) !
\
x6e
x
+∞
+∞
( −1) x 2 n = 1 − x 2 + x 4 − x 6 + ...
∑
2 24 720
( 2n ) !
n =0
n
+∞
( −1) x 2 n+1 = x − x3 + x5 − x 7 + ...
∑
6 120 5040
n = 0 ( 2n + 1) !
n
+∞
x 6 cos x
x 6 sin x
x 6 (1 + x )
α
Où α ∈ \
1
1− x
1
x6
1 − x2
1
x6
1+ x
1
x6
1 + x2
+∞
α (α − 1)(α − 2 ) ... (α − ( n − 1) )
n =1
n!
1+ ∑
+∞
∑x
x6
x6
x6
1
1 − x2
1
1 + x2
PanaMaths
xn
= 1 + x 2 + x 4 + x 6 + ...
]−1; +1[
x n = 1 − x + x 2 − x3 + ...
]−1; +1[
x 2 n = 1 − x 2 + x 4 − x 6 + ...
]−1; +1[
2n
n =0
∑ ( −1)
n
n =0
+∞
∑ ( −1)
n
]−1; +1[
]−1; +1[
+∞
+∞
\
= 1 + x + x 2 + x3 + ...
n
n =0
∑x
\
n =0
( 2n )! x 2 n = 1 + x 2 + 3x 4 + 5 x 6 + ...
∑
2
2n
2
8
16
n = 0 2 ( n !)
n
+∞
( −1) ( 2n )! x 2 n = 1 − x 2 + 3x 4 − 5 x 6 + ...
∑
2
2n
2
8
16
n = 0 2 ( n !)
+∞
[1-2]
]−1; +1[
]−1; +1[
Janvier 2010
+∞
x 6 ln (1 − x )
xn
x 2 x3 x 4
= − x − − − − ...
2 3 4
n
]−1; +1[
x 2 n +1
x3 x5 x 7
=
1
+
+ + + ...
∑
3 5 7
n = 0 2n + 1
]−1; +1[
−∑
n =1
+∞
x 6 arg tanh x
+∞
x 6 ln (1 + x )
∑
( −1)
n
n =1
+∞
∑ ( −1)
x 6 arctan x
n =0
n +1
n
xn
= x−
x 2 x3 x 4
+ − + ...
2 3 4
x 2 n +1
x3 x5 x 7
= x − + − + ...
2n + 1
3 5 7
( 2n )! x 2 n+1 = x + x3 + 3x5 + 5 x 7 + ...
2
2n
6 40 112
n=0
( n !) 2n + 1
x3 3 x5
π +∞ ( 2n ) ! x 2 n +1 π
−∑
= −x− −
− ...
2 n =0 22 n ( n !)2 2n + 1 2
6 40
n
+∞
( −1) ( 2n )! x 2 n+1 = x − x3 + 3x5 − 5 x 7 + ...
∑
2
2n
2n + 1
6 40 112
n = 0 2 ( n !)
+∞
x 6 arcsin x
x 6 arccos x
x 6 arg sinh x
PanaMaths
∑2
[2-2]
]−1; +1[
]−1; +1[
]−1; +1[
]−1; +1[
]−1; +1[
Janvier 2010