© Laurent Garcin MP Dumont d’Urville
Séries
Développements en série entière usuels
∀𝑧 ∈ ℂ, 𝑒𝑧=
+∞
∑
𝑛=0
𝑧𝑛
𝑛! ∀𝑥 ∈ ℝ, ch(𝑥) =
+∞
∑
𝑛=0
𝑥2𝑛 (2𝑛)!
∀𝑥 ∈ ℝ, sh(𝑥) =
+∞
∑
𝑛=0
𝑥2𝑛+1
(2𝑛 + 1)! ∀𝑥 ∈ ℝ, cos(𝑥) =
+∞
∑
𝑛=0
(−1)𝑛𝑥2𝑛 (2𝑛)!
∀𝑥 ∈ ℝ, sin(𝑥) =+∞∑
𝑛=0
(−1)𝑛𝑥2𝑛+1
(2𝑛 + 1)! ∀𝑥 ∈] − 1, 1[, arctan(𝑥) =+∞∑
𝑛=0
(−1)𝑛𝑥2𝑛+1 2𝑛 + 1
∀𝑧 ∈ ℂ, |𝑧| < 1 ⟹ 1 1 − 𝑧 =
+∞
∑
𝑛=0
𝑧𝑛 ∀𝑥 ∈] − 1, 1[, ln(1 + 𝑥) =
+∞
∑
𝑛=1
(−1)𝑛−1𝑥𝑛 𝑛
∀𝑥 ∈] − 1, 1[, (1 + 𝑥)α=
+∞
∑
𝑛=0
(α
𝑛)𝑥𝑛 avec(α
𝑛) = 1 𝑛!
𝑛−1
∏
𝑘=0
(α − 𝑘)
http://lgarcin.github.io
