Finding the asymptotic expansion of $\sin(3x)$ using asymptotic sequence $\{\ln(1+x^n)\}_n$. In the notes and lectures the only example that was given was an expansion for $\tan(x)$ where she literally just swapped the $x$ terms for $sin x$ terms since they are both asymptotic sequences. Is that the case here? Do I just find the regular expansion for $\sin(3x)$ and substitute in the relevant terms from the sequence?
Sorry if this is being confusing. Is this a solution? Since: $$\sin(3x)=(3x)-(3x)^3/6+...$$ and $$\ln(1+x)=x-1/2 x^2+1/3x^3-...,$$ $$\ln(1+x^2)=x^2-1/2x^4+...$$ $$\ln(1+x^3)=x^3$$ We have clearly that $a_1=3$, then $-1/2a_1+a_2=0$ and $1/3a_1+a_3=0$ So: $$\sin(3x)=3\ln(1+x)+3/2\ln(1+x^2)-\ln(1+x^3)$$?