<s>[INST] Expand a function in Maclaurin's series. The function is given with:
$$\ln(5\cos^{3}(x))$$ 
Need to be expanded:
$$x^{4}$$
I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the definition but I don't understand the concept enough to be able to solve problems like this.
 [/INST] $$\cos x=1-\frac{x^2}{2!}-\frac{x^4}{4!}+o(x^4)$$
and thus
$$5\cos^3(x)=5\left(1-\frac{x^2}{2!}+o(x^4)\right)^3=5-\frac{15x^2}{2!}+\frac{35}{8}x^4+o(x^4).$$
Moreover, if $x\longrightarrow  0$ you have that $5\cos^3(x)\longrightarrow  5$ and thus you have to develop $x\mapsto \ln(x)$ around $x=5$. That gives
$$\ln(x)=\ln(5)+\frac{1}{5}(x-5)-\frac{1}{25\cdot 2}(x-5)^2+o((x-5)^2).$$
Therefore
$$\ln(5\cos^3(x))=\ln(5)+\frac{1}{5}\left(-\frac{15x^2}{2!}+\frac{35}{8}x^4\right)-\frac{1}{50}\left(-\frac{15x^2}{2!}\right)^2+o(x^4)$$$$=\ln(5)-\frac{3}{2}x^2-\frac{1}{4}x^4+o(x^4)$$
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