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! 

A slight variant that requires no explicit cancelling of $1-b$ (so you needn't handle the $b=1$ case with a continuity argument) rearranges the double summation, viz. $$\sum_{n=0}^\infty\left(a^n\sum_{k=0}^n b^k\right)=\sum_{k=0}^\infty b^k\sum_{n=k}^\infty a^n=\sum_{k=0}^\infty b^ka^k\frac{1}{1-a}=\frac{1}{\left( 1-a\right)\left( 1-ab\right)}$$ provided $\left| a\right| < 1$ and $\left| ab\right| <1$.
