How to compute the sum $ 1+a(1+b)+a^2(1+b+b^2)+a^3(1+b+b^2+b^3)+\cdots$ Could it be possible to find the solution for the following series?    
$$ 1+a(1+b)+a^2(1+b+b^2)+a^3(1+b+b^2+b^3)+\cdots$$
Thanks in advance! 

Since the OP admitted that they can't continue from Dr.MV's hint, I'll do it myself:
$$\sum_{n=0}^\infty a^n \frac{1-b^{n+1}}{1-b}=\frac{1}{1-b} \left(\sum_{n=0}^\infty a^n-b \sum_{n=0}^\infty (ab)^n \right)=$$
Here we use geometric series, so we need to have $|a|<1$ and $|ab|<1$: *
$$=\frac{1}{1-b} \left(\frac{1}{1-a}-\frac{b}{1-ab} \right)=\frac{1}{(1-a)(1-ab)}$$
* Correction due to Batominovski's comment
