Table Dévelopements limités usuels

0
f(x) = f(0) + f(1)(0)x+f(2)(0) 1
2!x2+f(3)(0) 1 3!x3+··· +f(n)(0) 1 n!xn+o(xn).
f n f x = 0
f(x)
g(x)lim
x→0g(x)6= 0 DLnf(x)
DLng(x)x n
f(x) = DF (x)
dx DLnf(x)DLn+1F(x)
sinx x −1
3! x3+1 5! x5+··· + (−1)n1
(2n+1)! x2n+1 +o(x2n+1)
cosx 1−1
2! x2+1 4! x4+··· + (−1)n1
(2n)! x2n+o(x2n)
tanx =sin x
cos xx+1 3x3+2 15 x5+17 315 x7+··· +o(x2n+1)
ex1 + 1
1! x+1 2! x2+··· +1 n!xn+o(xn)
chx 1 + 1
2! x2+1 4! x4+··· +1
(2n)! x2n+o(x2n)
shx x +1
3! x3+1 5! x5+··· +1
(2n+1)! x2n+1 +o(x2n+1)
thx x −1
3x3+2 15 x5−17 315 x7+··· +o(x2n+1)
1
1+x1−x+x2−x3+··· + (−1)nxn+o(xn)
1
1−x1 + x+x2+x3+··· +xn+o(xn)
ln(1 + x)x−1
2x2+1 3x3−1 4x4+··· + (−1)n−11 nxn+o(xn)
DLnn.
DLn+1 n+ 1.

f n f x = 0
(1 + x)α, α ∈R1 + αx +α(α−1)
2! x2+··· +α(α−1)···(α−(n−1)) n!xn+o(xn)
√1 + x, α =1
21 + 1 2x−1
22·2! x2+1·3 23·3! x3−1·3·5 24·4! x4+··· + (−1)n−11·3·5·7···(2n−3)
2n·n!xn+o(xn)
√1−x, α =1
2, x −→ −x1−1 2x−1
22·2! x2−1·3 23·3! x3−1·3·5 24·4! x4− ··· − 1·3·5·7···(2n−3)
2n·n!xn+o(xn)
1
√1+x, α =−1
21−1 2x+1.3
22
1
2! x2−1.3.5
23
1
3! x3+1.3.5.7
24
1
4! x4+··· + (−1)n1·3·5·7···(2n−1) 2nn!xn+o(xn)
1
√1−x, α =−1
2, x −→ −x1 + 1 2x+1·3
22
1
2! x2+1·3·5
23
1
3! x3+1·3·5·7
24
1
4! x4+··· +1·3·5·7···(2n−1) 2nn!xn+o(xn)
1
1+x2, α =−1, x −→ x21−x2+x4−x6+··· + (−1)nx2n+o(x2n+1)
1
1−x2, α =−1, x −→ −x21 + x2+x4+x6+··· +x2n+o(x2n+1)
1
√1+x2, α =−1
2, x −→ x21−1 2x2+1·3
22·2! x4−1·3·5 23·3! x6+1·3·5·7 24·4! x8+···+ (−1)n1·3·5·7···(2n−1)
2n·n!x2n+o(x2n+1)
1
√1−x2, α =−1
2, x −→ −x21 + 1 2x2+1·3
22·2! x4+1·3·5 23·3! x6+1·3·5·7 24·4! x8+··· +1·3·5·7·(2n−1)
2n·n!x2n+o(x2n+1)
arcsin x x +1
2·3x3+1·3 2·4·5x5+1·3·5 2·4·6·7x7+1·3·5·7 2·4·6·8·9x9+···+1·3·5···(2n−1)
2·4···(2n)·(2n+1) x2n+1 +o(x2n+2)
arccos xπ
2−arcsinx =π 2−x−1 2·3x3−1·3 2·4·5x5− ··· − 1·3·5···(2n−1)
2·4···(2n)·(2n+1) x2n+1 +o(x2n+2)
arctanx x −1
3x3+1 5x5−1 7x7+··· + (−1)n1 2n+1 x2n+1 +o(x2n+2)
argshx x −1
2·3x3+1·3 2·4·5x5−1·3·5 2·4·6·7x7+··· + (−1)n1·3·5···(2n−1)
2·4···(2n)·(2n+1) x2n+1 +o(x2n+2)
argthx x +1
3x3+1 5x5+1 7x7+··· +1 2n+1 x2n+1 +o(x2n+2)

f
arcsinx 1
√1−x2]−1,1[
arccosx −1
√1−x2]−1,1[
arctanx 1
1+x2R
chx shx R
shx chx R
thx 1−th2x=1
ch2xR
argshx 1
√1+x2R
argchx 1
√x2−1]1,+∞[
argthx 1
1−x2]−1,1[