While working on a probability problem I have found that the following series should be finite: $$ \displaystyle\sum_{i=1}^\infty\left\lbrace[\frac{1}{c}(1-q^i)-1]^m+1\right\rbrace $$ (for $m$ an odd positive integer and $c$ a positive integer major or equal than two). And: $$ \displaystyle\sum_{i=1}^\infty \left\lbrace -[\frac{1}{c}(1-q^i)-1]^m+1\right\rbrace $$ for $m$ an even positive integer and $c$ a positive integer major or equal than two.
Sadly, I've not been able to prove both formulae are finite.