Ce formulaire de développements en série recense des développements en série de fonctions pour les fonctions de référence (pour la plupart, des séries entières , et quelques séries de Laurent ). Elles sont données avec indication du domaine de convergence (le rayon de convergence pour les séries entières) dans le champ complexe ou réel. La notation
D
(
a
,
r
)
{\displaystyle D(a,r)}
C
{\displaystyle \mathbb {C} }
a
{\displaystyle a}
r
{\displaystyle r}
B
n
{\displaystyle B_{n}}
n -ième nombre de Bernoulli .
∀
x
∈
D
(
0
,
1
)
,
1
1
−
x
=
∑
n
=
0
+
∞
x
n
.
{\displaystyle \forall x\in D(0,1),\,{1 \over {1-x}}=\sum _{n=0}^{+{\infty }}{x^{n}}.}
∀
x
∈
]
−
1
,
1
[
,
∀
α
∈
R
,
(
1
+
x
)
α
=
1
+
∑
n
=
1
+
∞
α
(
α
−
1
)
…
(
α
−
n
+
1
)
n
!
x
n
.
{\displaystyle \forall x\in \,]-1,1[,\ \forall \alpha \in \mathbb {R} ,\,(1+x)^{\alpha }\,=1\;+\;\sum _{n=1}^{+{\infty }}{{\frac {\alpha \,(\alpha -1)\ldots (\alpha -n+1)}{n!}}\,x^{n}}.}
[ 1] En particulier :
∀
x
∈
R
,
∀
m
∈
N
,
(
1
+
x
)
m
=
1
+
∑
n
=
1
m
m
(
m
−
1
)
…
(
m
−
n
+
1
)
n
!
x
n
=
∑
n
=
0
m
(
m
n
)
x
n
.
{\displaystyle \forall x\in \mathbb {R} ,\,\forall m\in \mathbb {N} ,\,(1+x)^{m}\,=1\;+\;\sum _{n=1}^{m}{{\frac {m\,(m-1)\ldots (m-n+1)}{n!}}\,x^{n}}=\sum _{n=0}^{m}{{m \choose n}\,x^{n}}.}
∀
x
∈
[
−
1
,
1
]
,
1
+
x
=
∑
n
=
0
+
∞
(
−
1
)
n
+
1
2
n
−
1
(
2
n
n
)
2
2
n
x
n
.
{\displaystyle \forall x\in \,[-1,1],\ {\sqrt {1+x}}\,=\sum _{n=0}^{+{\infty }}{{\frac {(-1)^{n+1}}{2n-1}}{\frac {\binom {2n}{n}}{2^{2n}}}\,x^{n}}.}
∀
x
∈
]
−
1
,
1
[
,
∀
α
∈
R
,
1
(
1
−
x
)
α
=
1
+
∑
n
=
1
+
∞
α
(
α
+
1
)
…
(
α
+
n
−
1
)
n
!
x
n
.
{\displaystyle \forall x\in \,]-1,1[,\ \forall \alpha \in \mathbb {R} ,\,{\frac {1}{(1-x)^{\alpha }}}\,=1\;+\;\sum _{n=1}^{+{\infty }}{{\frac {\alpha \,(\alpha +1)\ldots (\alpha +n-1)}{n!}}\,x^{n}}.}
∀
x
∈
]
−
1
,
1
[
,
∀
m
∈
N
∗
,
1
(
1
−
x
)
m
=
∑
n
=
0
+
∞
(
m
+
n
−
1
n
)
x
n
{\displaystyle \forall x\in \,]-1,1[,\ \forall m\in \mathbb {N} ^{*},\,{\frac {1}{(1-x)^{m}}}\,=\sum _{n=0}^{+{\infty }}{{\binom {m+n-1}{n}}\,x^{n}}}
formule du binôme négatif ).
∀
x
∈
[
−
1
,
1
[
,
1
1
−
x
=
∑
n
=
0
+
∞
(
2
n
n
)
2
2
n
x
n
.
{\displaystyle \forall x\in \,[-1,1[,\ {\frac {1}{\sqrt {1-x}}}\,=\sum _{n=0}^{+{\infty }}{{\frac {\binom {2n}{n}}{2^{2n}}}\,x^{n}}.}
Pour tout nombre complexe z réel a > 0
e
z
=
∑
n
=
0
+
∞
z
n
n
!
=
1
+
z
1
!
+
z
2
2
!
+
z
3
3
!
+
z
4
4
!
+
⋯
{\displaystyle {\rm {e}}^{z}=\sum _{n=0}^{+\infty }{\frac {z^{n}}{n!}}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+{\frac {z^{4}}{4!}}+\cdots }
a
z
=
e
z
ln
a
=
∑
n
=
0
+
∞
(
ln
a
)
n
n
!
z
n
=
1
+
ln
a
1
!
z
+
(
ln
a
)
2
2
!
z
2
+
(
ln
a
)
3
3
!
z
3
+
(
ln
a
)
4
4
!
z
4
+
⋯
{\displaystyle a^{z}={\rm {e}}^{z\ln a}=\sum _{n=0}^{+\infty }{\frac {(\ln a)^{n}}{n!}}{z^{n}}=1+{\frac {\ln a}{1!}}z+{\frac {(\ln a)^{2}}{2!}}z^{2}+{\frac {(\ln a)^{3}}{3!}}z^{3}+{\frac {(\ln a)^{4}}{4!}}z^{4}+\cdots }
|
z
|
≤
1
et
z
≠
−
1
⇒
ln
(
1
+
z
)
=
∑
n
=
1
+
∞
(
−
1
)
n
+
1
z
n
n
=
z
−
z
2
2
+
z
3
3
−
z
4
4
+
⋯
{\displaystyle |z|\leq 1{\text{ et }}z\neq -1\Rightarrow \ln(1+z)=\sum _{n=1}^{+\infty }{(-1)^{n+1}{\frac {z^{n}}{n}}}=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-{\frac {z^{4}}{4}}+\cdots }
[ 1]
∀
x
∈
C
,
sin
x
=
∑
n
=
0
+
∞
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
=
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
{\displaystyle \forall x\in \mathbb {C} ,\,\sin x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2n+1}}{(2\,n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }
∀
x
∈
C
,
cos
x
=
∑
n
=
0
+
∞
(
−
1
)
n
x
2
n
(
2
n
)
!
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
⋯
{\displaystyle \forall x\in \mathbb {C} ,\,\cos x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2n}}{(2\,n)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }
∀
x
∈
D
(
0
,
π
2
)
,
tan
x
=
∑
n
=
1
+
∞
(
−
1
)
n
−
1
2
2
n
(
2
2
n
−
1
)
(
2
n
)
!
B
2
n
x
2
n
−
1
=
x
+
1
3
x
3
+
2
15
x
5
+
17
315
x
7
+
62
2835
x
9
+
⋯
{\displaystyle \forall x\in D(0,{\frac {\pi }{2}}),\tan x=\sum _{n=1}^{+\infty }(-1)^{n-1}\,{\frac {2^{2n}(2^{2n}-1)}{(2n)!}}B_{2n}x^{2n-1}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+{\frac {62}{2835}}x^{9}+\cdots }
∀
x
∈
]
−
π
2
,
π
2
[
,
tan
x
=
2
π
∑
n
=
0
+
∞
(
x
π
)
2
n
+
1
(
2
2
n
+
2
−
1
)
ζ
(
2
n
+
2
)
avec
∀
p
>
1
,
ζ
(
2
p
)
=
∑
n
=
1
+
∞
1
n
2
p
=
|
B
2
p
|
(
2
π
)
2
p
2
(
2
p
)
!
{\displaystyle \forall x\in \,\left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[,\,\tan x={\frac {2}{\pi }}\,\sum _{n=0}^{+{\infty }}\,{\left({\frac {x}{\pi }}\right)}^{2\,n+1}(2^{2\,n+2}-1)\;\zeta (2\,n+2)\quad {\text{avec}}\;\forall p>1,\,\zeta (2p)=\sum _{n=1}^{+{\infty }}\,{\frac {1}{n^{2p}}}={\frac {|B_{2p}|(2\pi )^{2p}}{2(2p)!}}}
ζ
{\displaystyle \zeta }
fonction zêta de Riemann et les
B
k
{\displaystyle B_{k}}
nombres de Bernoulli .
∀
x
∈
D
(
0
,
π
)
∖
{
0
}
,
cot
x
=
1
x
−
∑
n
=
1
+
∞
2
2
n
(
2
n
)
!
B
2
n
x
2
n
−
1
=
1
x
−
x
3
−
x
3
45
−
2
x
5
945
⋯
{\displaystyle \forall x\in D(0,\pi )\setminus \lbrace 0\rbrace ,\cot x={\frac {1}{x}}-\sum _{n=1}^{+\infty }{\frac {2^{2n}}{(2n)!}}B_{2n}x^{2n-1}={\frac {1}{x}}-{\frac {x}{3}}-{\frac {x^{3}}{45}}-{\frac {2x^{5}}{945}}\cdots }
∀
x
∈
[
−
1
,
1
]
,
arcsin
x
=
∑
n
=
0
+
∞
(
2
n
n
)
2
2
n
×
x
2
n
+
1
2
n
+
1
=
x
+
1
2
⋅
3
x
3
+
1
⋅
3
2
⋅
4
⋅
5
x
5
+
1
⋅
3
⋅
5
2
⋅
4
⋅
6
⋅
7
x
7
+
.
.
.
{\displaystyle \forall x\in [-1,1],\,\arcsin x=\sum _{n=0}^{+\infty }{\frac {\binom {2n}{n}}{2^{2n}}}\times {\frac {x^{2n+1}}{2n+1}}=x+{\frac {1}{2\cdot 3}}x^{3}+{\frac {1\cdot 3}{2\cdot 4\cdot 5}}x^{5}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}}x^{7}+...}
[ 1]
∀
x
∈
[
−
1
,
1
]
,
arccos
x
=
π
2
−
arcsin
x
=
π
2
−
∑
n
=
0
+
∞
(
2
n
n
)
2
2
n
×
x
2
n
+
1
2
n
+
1
=
π
2
−
(
x
+
1
2
⋅
3
x
3
+
1
⋅
3
2
⋅
4
⋅
5
x
5
+
1
⋅
3
⋅
5
2
⋅
4
⋅
6
⋅
7
x
7
+
⋯
)
{\displaystyle \forall x\in [-1,1],\,\arccos x={\frac {\pi }{2}}-\arcsin x={\frac {\pi }{2}}-\sum _{n=0}^{+\infty }{\frac {\binom {2n}{n}}{2^{2n}}}\times {\frac {x^{2n+1}}{2n+1}}={\frac {\pi }{2}}-\left(x+{\frac {1}{2\cdot 3}}x^{3}+{\frac {1\cdot 3}{2\cdot 4\cdot 5}}x^{5}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}}x^{7}+\cdots \right)}
[ 1]
∀
x
∈
[
−
1
,
1
]
,
arctan
x
=
∑
n
=
0
+
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
=
x
−
x
3
3
+
x
5
5
−
x
7
7
+
⋯
{\displaystyle \forall x\in [-1,1],\arctan x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2\,n+1}}{2\,n+1}}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots }
[ 1]
x
=
1
{\displaystyle x=1}
π
=
4
∑
n
=
0
+
∞
(
−
1
)
n
2
n
+
1
{\displaystyle \pi =4\,\sum _{n=0}^{+{\infty }}{\frac {(-1)^{n}}{2\,n+1}}}
∀
x
∈
[
−
1
,
1
]
,
arccot
x
=
π
2
−
arctan
x
=
π
2
−
∑
n
=
0
+
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
=
π
2
−
(
x
−
x
3
3
+
x
5
5
−
x
7
7
+
⋯
)
{\displaystyle \forall x\in [-1,1],\operatorname {arccot} x={\frac {\pi }{2}}-\arctan x={\frac {\pi }{2}}-\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2\,n+1}}{2\,n+1}}={\frac {\pi }{2}}-\left(x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \right)}
∀
x
∈
[
−
1
,
1
]
,
arcsin
2
x
=
1
2
∑
n
=
1
∞
(
2
x
)
2
n
n
2
(
2
n
n
)
=
x
2
+
x
4
3
+
8
x
6
45
+
⋯
{\displaystyle \forall x\in [-1,1],\arcsin ^{2}x={\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {(2x)^{2n}}{n^{2}{\binom {2n}{n}}}}=x^{2}+{\frac {x^{4}}{3}}+{\frac {8x^{6}}{45}}+\cdots }
[ 2] , [ 3]
∀
x
∈
C
,
sinh
x
=
∑
n
=
0
+
∞
x
2
n
+
1
(
2
n
+
1
)
!
=
x
+
x
3
3
!
+
x
5
5
!
+
x
7
7
!
+
⋯
{\displaystyle \forall x\in \mathbb {C} ,\,\sinh x=\sum _{n=0}^{+\infty }{\frac {x^{2n+1}}{(2\,n+1)!}}=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots }
∀
x
∈
C
,
cosh
x
=
∑
n
=
0
+
∞
x
2
n
(
2
n
)
!
=
1
+
x
2
2
!
+
x
4
4
!
+
x
6
6
!
+
⋯
{\displaystyle \forall x\in \mathbb {C} ,\,\cosh x=\sum _{n=0}^{+{\infty }}{\frac {x^{2n}}{(2\,n)!}}=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots }
∀
x
∈
]
−
π
2
,
π
2
[
,
tanh
x
=
∑
n
=
1
+
∞
2
2
n
(
2
2
n
−
1
)
(
2
n
)
!
B
2
n
x
2
n
−
1
=
x
−
1
3
x
3
+
2
15
x
5
−
17
315
x
7
+
⋯
{\displaystyle \forall x\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[,\operatorname {tanh} \,x=\sum _{n=1}^{+\infty }{\frac {2^{2n}(2^{2n}-1)}{(2n)!}}B_{2n}x^{2n-1}=x-{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}-{\frac {17}{315}}x^{7}+\cdots }
∀
x
∈
]
0
,
π
[
,
coth
x
=
1
x
+
∑
n
=
1
+
∞
2
2
n
(
2
n
)
!
B
2
n
x
2
n
−
1
=
1
x
+
1
3
x
−
1
45
x
3
+
2
945
x
5
−
⋯
{\displaystyle \forall x\in ]0,\pi [,\coth x={\frac {1}{x}}+\sum _{n=1}^{+\infty }{\frac {2^{2n}}{(2n)!}}B_{2n}x^{2n-1}={\frac {1}{x}}+{\frac {1}{3}}x-{\frac {1}{45}}x^{3}+{\frac {2}{945}}x^{5}-\cdots }
∀
x
∈
[
−
1
,
1
]
,
arsinh
x
=
ln
(
x
+
x
2
+
1
)
=
x
+
∑
n
=
1
+
∞
(
−
1
)
n
(
2
n
n
)
2
2
n
×
x
2
n
+
1
2
n
+
1
=
x
−
1
2
⋅
3
x
3
+
1
⋅
3
2
⋅
4
⋅
5
x
5
−
1
⋅
3
⋅
5
2
⋅
4
⋅
6
⋅
7
x
7
+
⋯
{\displaystyle \forall x\in [-1,1],\,\operatorname {arsinh} \,x=\ln \left(x+{\sqrt {x^{2}+1}}\right)=x+\sum _{n=1}^{+{\infty }}\,(-1)^{n}{\frac {\binom {2n}{n}}{2^{2n}}}\times {\frac {x^{2n+1}}{2\,n+1}}=x-{\frac {1}{2\cdot 3}}x^{3}+{\frac {1\cdot 3}{2\cdot 4\cdot 5}}x^{5}-{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 7}}x^{7}+\cdots }
∀
x
∈
[
1
,
+
∞
[
,
arcosh
x
=
ln
(
x
+
x
2
−
1
)
=
ln
(
2
x
)
−
∑
n
=
1
+
∞
(
2
n
n
)
2
2
n
×
1
2
n
×
x
2
n
=
ln
(
2
x
)
−
1
2
⋅
2
x
2
−
1
⋅
3
2
⋅
4
⋅
4
x
4
−
1
⋅
3
⋅
5
2
⋅
4
⋅
6
⋅
6
x
6
−
⋯
{\displaystyle \forall x\in [1,+\infty [,\,\operatorname {arcosh} \,x=\ln \left(x+{\sqrt {x^{2}-1}}\right)=\ln(2x)-\sum _{n=1}^{+\infty }\,{\frac {\binom {2n}{n}}{2^{2n}}}\times {\frac {1}{2n\times x^{2n}}}=\ln(2x)-{\frac {1}{2\cdot 2x^{2}}}-{\frac {1\cdot 3}{2\cdot 4\cdot 4x^{4}}}-{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 6x^{6}}}-\cdots }
∀
x
∈
]
−
1
,
1
[
,
artanh
x
=
1
2
ln
(
1
+
x
1
−
x
)
=
∑
n
=
0
+
∞
x
2
n
+
1
2
n
+
1
=
x
+
x
3
3
+
x
5
5
+
x
7
7
+
⋯
{\displaystyle \forall x\in ]-1,1[,\,\operatorname {artanh} \,x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)=\sum _{n=0}^{+{\infty }}\,{\frac {x^{2n+1}}{2n+1}}=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots }
∀
x
∈
]
−
∞
,
1
[
∪
]
1
,
+
∞
[
,
arcoth
x
=
1
2
ln
(
x
+
1
x
−
1
)
=
∑
n
=
0
+
∞
1
(
2
n
+
1
)
×
x
2
n
+
1
=
1
x
+
1
3
x
3
+
1
5
x
5
+
1
7
x
7
+
⋯
{\displaystyle \forall x\in ]-\infty ,1[\cup ]1,+\infty [,\,\operatorname {arcoth} \,x={\frac {1}{2}}\ln \left({\frac {x+1}{x-1}}\right)=\sum _{n=0}^{+{\infty }}\,{\frac {1}{(2n+1)\times x^{2n+1}}}={\frac {1}{x}}+{\frac {1}{3x^{3}}}+{\frac {1}{5x^{5}}}+{\frac {1}{7x^{7}}}+\cdots }
∀
x
∈
[
−
1
,
1
]
,
arsinh
2
x
=
∑
n
=
1
∞
(
−
1
)
n
+
1
(
2
x
)
2
n
n
2
(
2
n
n
)
=
x
2
−
x
4
3
+
8
x
6
45
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{\displaystyle \forall x\in [-1,1],{\text{arsinh}}^{2}x=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(2x)^{2n}}{n^{2}{\binom {2n}{n}}}}=x^{2}-{\frac {x^{4}}{3}}+{\frac {8x^{6}}{45}}+\cdots }
Valentin Strach , « Les développements en série entière usuels Progresser-en-maths , 24 novembre 2021 (consulté le 11 juillet 2025 ) ↑ (en) Karl Dilcher et Christophe Vignat, « Powers of the arcsine and infinite classes of series involving central binomial coefficients ↑ (en) D. H. Lehmer , « Interesting Series Involving the Central Binomial Coefficient », The American Mathematical Monthly vol. 92, no 7, aug.-sep. 1985, p. 449-457 (JSTOR 2322496
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