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

Mar 11, 2015

Hallo,

• First, you have to know the usual serie :
$\ln \left(1 + X\right) = X - {X}^{2} / 2 + {X}^{3} / 3 + \setminus \ldots + {\left(- 1\right)}^{n - 1} {X}^{n} / n + {o}_{X \to 0} \left({X}^{n}\right)$

• Second, you can take $X = {x}^{3}$ because if $x \to 0$, then ${x}^{3} \to 0$. So,
$\ln \left(1 + {x}^{3}\right) = {x}^{3} - {x}^{6} / 2 + {x}^{9} / 3 + \setminus \ldots + {\left(- 1\right)}^{n - 1} {x}^{3 n} / n + {o}_{x \to 0} \left({x}^{3 n}\right)$

• Finally, multiply by $2 x$ :
$2 x \ln \left(1 + {x}^{3}\right) = 2 {x}^{4} - {x}^{7} + \frac{2}{3} {x}^{10} + \setminus \ldots + {\left(- 1\right)}^{n - 1} \frac{2}{n} {x}^{3 n + 1} + {o}_{x \to 0} \left({x}^{3 n + 1}\right)$