How do I obtain the Maclaurin series for $f(x)= 2xln(1+x3)$ ?

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Answer 1 · 2015-03-11T21:03:16

Hallo,

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 $2x$ :
$2x ln(1+x^3) = 2x^4 - x^7 + 2/3 x^10+ \ldots + (-1)^(n-1)2/n x^{3n+1} + o_{x->0}(x^(3n+1))$

How do I obtain the Maclaurin series for f(x)= 2xln(1+x3)f(x)= 2xln(1+x3)?