correction_dl.py

Created on March 03, 2022
818 Bytes
# DL ordre 3 et x->0

# sin(ln(1+x))

# X=ln(1+x)    lim X=0
#              x->0
# sinX= x- x^3/6 + o(x^3)

# X=x- x^2/2 + x^3/3 + o(x^3)  
# X^3= X^2*X = (x- x^2/2 + x^3/3 +o(x^3))^2 * (x- x^2/2 + x^3/3 +o(x^3)) 
# X^3= (x^2 - x^3 + o(x^3))(x - x^2/2 + x^3/3 +o(x^3))
# X^3= x^3 +o(x^3)
# sinX = x- x^2/2 + x^3/3 + o(x^3) - 1/6 *x^3 +o(x^3)
# sin(ln(1+x))=x- x^2/2 + 1/6 *x^3 +o(x^3)



# e^2x * ln(1+x)
# e^2x = 1 + 2x + 1/2 *(2x)^2 + 1/6 *(2x)^3 +o(x^3)
#      = 1 + 2x + 2x^2 + 4/3 *x^3 +o(x^3)
# ln(1+x) = x - x^2/2 + x^3/3 + o(x^3)
# e^2x * ln(1+x) = (1+2x+2x^2 + 4/3 * x^3 +o(x^3))(x- x^2/2 +x^3/3 + o(x^3))
# = x- x^2/2 + x^3/3 +2x^2 - x^3 + 2x^3 +o(x^3)
# = x + 3/2 *x^2 + 3/4 *x^3 + o(x^3)
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