I'm looking for merging this summations into a single one:
$\displaystyle \sum_{i=0}^k{\sum_{j=0}^i{b^j}}$ where $b$ is an integer.
I know it is equal to: $\displaystyle \sum_{i=0}^k{b^i(k-i+1)}$
In fact, one can see that: $\displaystyle \sum_{i=0}^k{\sum_{j=0}^i{b^j}} = (b^0) + (b^0 + b^1) + (b^0 + b^1 + b^2) + ... + (b^0 + b^1 + b^2 + ... + b^k)$, where any $b^j$ appears from $i=j$ to $i=k$ in such a way that it can be found $k-j+1$ times.
However, I would like to know a more rigorous demonstration, i.e. in a more "closed" form, without involving these ellipsis.
