How do you obtain the Maclaurin series for #f(x)= x^2ln(1+x^3)#?

1 Answer
Mar 11, 2015

Hi,

  • First, you have to know the usual serie :
    #ln(1+X) = X - X^2/2 + X^3/3 + \ldots + (-1)^(n-1)X^n/n + o_{X->0}(X^n)#

  • Second, you can take #X=x^3# because if #x->0#, then #x^3->0#. So,
    #ln(1+x^3) = x^3 - x^6/2 + x^9/3+\ldots + (-1)^(n-1)x^{3n}/n + o_{x->0}(x^(3n))#

  • Finally, multiply by #2x# :
    #2x ln(1+x^3) = 2x^4 - x^7 + 2/3 x^10+ \ldots + (-1)^(n-1)2/n x^{3n+1} + o_{x->0}(x^(3n+1))#