f tq f(n) soit continue au voisinage de 0

 

MacLaurin avec reste d'Young

f(x) = f(0) + xf'(0) + x²/2 f''(0) + … + xn/n! f(n)(0) + xn ε(x)

où ε(x) ® 0 quand x®0

 

sin(x) = x – x3/3! + x5/5! – … + (-1)n x2n+1/(2n+1)! + x2n+2 ε(x)

 

cos(x) = 1 – x2/2! + x4/4! – … + (-1)n x2n/(2n)! + x2n+2 ε(x)

 

ex = 1 + x/1 + x²/2 + … + xn/n! + xn ε(x)

 

1/(1-x) = 1 + x + x² + … + xn + xn ε(x)

 

1/(1+x) = 1 – x + x² – … + (-1)n xn + xn ε(x)

 

ln(1+x) = x – x²/2 + … + (-1)n+1 xn/n + xn ε(x)

 

1/(1+x²) = 1 – x² + x4 – x6 + … + (-1)n x2n + x2n+1 ε(x)

 

arctan(x) = x – x3/3 + x5/5 – … + (-1)n x2n+1 / (2n+1) + x2n+2 ε(x)

 

(1+x)a = 1 + ax + a(a-1)x²/2! + a(a-1)(a-2) x3/3! + … +

a(a-1)…(a-n+1) xn/n! + xn ε(x)

 

√(1+x) = 1 + x/2 – x²/8 + x3/16 – 5x4/128 + x4 ε(x)

 

tan(x) = x + x3/3 + 2x5/15 + 17x7/315 + x8 ε(x)