<s>[INST] complex series summation up to infinity 
Sum  of series 
  $$S(x)=\frac{1}{1+x}+\frac{2x}{1+x^2}+\frac{3x^2}{1+x^3}+\frac{4x^3}{1+x^4}+\cdots , |x|<1$$

What I tried.
Let $$A=\ln\bigg[(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots \cdots \bigg]$$
$$A = \ln\bigg[(1+x+x^2+2x^3+2x^4+3x^5+\cdots+ )\bigg]$$
How do is solve it help me please 
 [/INST] Regarding the relationship with the $A$ term
$$
S_n(x) = \sum_{k=1}^n\frac{g_k'(x)}{g_k(x)} = \frac{d}{dx}\sum_{k=1}^n\ln g_k(x) = \frac{d}{dx}\ln\prod_{k=1}^ng_k(x) = \frac{d}{dx}A 
$$
here $g_k(x) = 1+x^k$.  Also $\prod_{k=1}^n g_k(x) = \frac 12(-1;x)_{n+1}$  (see) and 
$$
S_n(x) = \frac{d}{dx}\ln\left(\frac 12(-1;x)_{n+1}\right)
$$
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