<s>[INST] 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! 
 [/INST] HINT:
$$\sum_{n=0}^\infty a^n\sum_{m=0}^n b^m=\sum_{n=0}^\infty a^n \frac{1-b^{n+1}}{1-b}$$
Can you finish?
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