Computation of Maclaurin Series

Question

I have been working on Maclaurin Series recently and was wondering if there's a more simple and elegant way to obtain series for more complicated functions,say $f(x)=\ln(1+2x+2x^2)$ or $g(x)=\tan(2x^4-x)$.Using the definition leads to messy derivatives almost immediately.If it was some simple rational function,for example,i would try to use Maclaurin Series of ${1\over1+x}$ or ${1\over1-x}$ and then manipulate it to get my result,but I can't really think of any shortcut for listed above functions(and many more).

Answer 1

keep substitution in mind. It may not give the whole infinite series, but you will usually get the first several terms. So, $$ \frac{1}{1+t} = 1 - t + t^2 - t^3 + t^4 - t^5 \cdots $$ $$ \log (1+t) = t - \frac{t^2}{2} + \frac{t^3}{3} - \frac{t^4}{4} \cdots $$ Taking $t = 2x+2x^2$ correctly gives the first few terms of $\log(1+2x+2x^2),$ up to $x^4$

$$ \log(1+2x+2x^2) = 2 x - \frac{4x^3}{3} + 2 x^4 \cdots $$