<s>[INST] Taylor series of $(1+3x) \cdot \ln(1+x)$ I have to find Taylor series of $(1+3x) \cdot \ln(1+x)$. I know Taylor series of $(1+3x) \cdot \ln(1+x)$ but I do not know hot to simplify.
Any help?
 [/INST] The Maclaurin series of the given function is the product of the Taylor series of $\ln(1+x)$ and $1+3x$, treating the former as if it were a regular polynomial:
$$\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\dots$$
$$3x\ln(1+x)=3x^2-\frac{3x^3}2+\frac{3x^4}3-\frac{3x^5}4+\dots$$
$$(1+3x)\ln(1+x)=x+\left(3-\frac12\right)x^2-\left(\frac32-\frac13\right)x^3+\left(\frac33-\frac14\right)x^4-\dots$$
$$=x+\sum_{k=2}^\infty(-1)^k\left(\frac3{k-1}-\frac1k\right)x^k$$
$$=x+\sum_{k=2}^\infty(-1)^k\frac{2(k+1)}{k(k-1)}x^k$$
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