D´ eveloppements limit´ es
D´eveloppements limit´es, fa¸con Taylor-Young, au voisinage de 0 (`a connaˆıtre par cœur) : sin(x) =x− x3
3! + x5
5! +· · ·+ (−1)n x2n+1
(2n+ 1)! +x2n+1ε(x) cos(x) = 1−x2
2 +x4
4! +· · ·+ (−1)n x2n
(2n)!+x2nε(x) ln(1 +x) =x− x2
2 + x3
3 +· · ·+ (−1)nxn+1
n+ 1+xn+1ε(x) ex= 1 +x+x2
2 +x3
3! +· · ·+xn
n! +xnε(x) 1
1 +x = 1−x+x2−x3+· · ·+ (−1)nxn+xnε(x) (1 +x)a= 1 +ax+a(a−1)
2 x2+a(a−1)(a−2)
3! x3+· · ·+ a(a−1)(a−2)...(a−n+ 1)
n! xn+xnε(x) sinh(x) =x+ x3
3! + x5
5! +· · ·+ x2n+1
(2n+ 1)! +x2n+1ε(x) cosh(x) = 1 +x2
2 +x4
4! +· · ·+ x2n
(2n)!+x2nε(x)
D´eveloppements limit´es qu’on calcule ais´ement `a partir des pr´ec´edents : tan(x) =x+x3
3 +2x5
15 +x5ε(x) th(x) =x−x3
3 +2x5
15 +x5ε(x) arctan(x) =x−x3
3 +x5
5 +· · ·+ (−1)nx2n+1
2n+ 1+x2n+2ε(x) argth(x) =x+x3
3 +· · ·+ x2n+1
2n+ 1+x2n+2ε(x) arcsin(x) =x+1.x3
2.3 +1.3.x5
2.4.5 +· · ·+1.3.5....(2n−1)x2n+1
2.4.6....(2n)(2n+ 1) +x2n+2ε(x) argsh(x) =x−1.x3
2.3 +· · ·+ (−1)n1.3.5....(2n−1)x2n+1
2.4.6....(2n)(2n+ 1) +x2n+2ε(x) En physique, on utilise fr´equemment des d´eveloppements limit´es d’ordre 1 :
(1 +x)n= 1 +nx+xε(x)
√
1 +x= 1 +1
2x+xε(x) 1
1 +x = 1−x+xε(x) sinx=x+x2ε(x) tanx=x+x2ε(x)
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