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Développements en série entière usuels


P∞ an
eax = xn a ∈ C, x ∈ R
n=0 n!

P
∞ 1
sh x = x2n+1 x∈R
n=0 (2n + 1)!
P
∞ 1
ch x = x2n x∈R
n=0 (2n)!
P∞ (−1)n
sin x = x2n+1 x∈R
n=0 (2n + 1)!

P∞ (−1)n
cos x = x2n x∈R
n=0 (2n)!


P∞ α(α − 1) · · · (α − n + 1)
(1 + x)α = 1+ xn (α ∈ R) x ∈ ] −1 ; 1 [
n=1 n!
1 P
∞ 1
= n+1
xn (a ∈ C∗ ) x ∈ ] −|a| ; |a| [
a−x n=0 a
1 P∞ n+1
= n+2
xn (a ∈ C∗ ) x ∈ ] −|a| ; |a| [
(a − x)2 n=0 a

1 P∞ Ck−1
n+k−1
= n+k
xn (a ∈ C∗ ) x ∈ ] −|a| ; |a| [
(a − x)k n=0 a

P∞ 1
ln(1 − x) = − xn x ∈ [ −1 ; 1 [
n=1 n

P∞ (−1)n−1
ln(1 + x) = xn x ∈ ] −1 ; 1 ]
n=1 n
√ x P
∞ 1 × 3 × · · · × (2n − 3) n
1+x = 1+ + (−1)n−1 x x ∈ ] −1 ; 1 [
2 n=2 2 × 4 × · · · × (2n)
1 P
∞ 1 × 3 × · · · × (2n − 1) n
√ = 1+ (−1)n x x ∈ ] −1 ; 1 [
1+x n=1 2 × 4 × · · · × (2n)
P∞ (−1)n
Arctan x = x2n+1 x ∈ [ −1 ; 1 ]
n=0 2n + 1

P
∞ 1
Argth x = x2n+1 x ∈ ] −1 ; 1 [
n=0 2n + 1
P∞ 1 × 3 × · · · × (2n − 1) x2n+1
Arcsin x = x+ x ∈ ] −1 ; 1 [
n=1 2 × 4 × · · · × (2n) 2n + 1
P
∞ 1 × 3 × · · · × (2n − 1) x2n+1
Argsh x = x+ (−1)n x ∈ ] −1 ; 1 [
n=1 2 × 4 × · · · × (2n) 2n + 1