I have found that if $$ \begin{array}{ll} &u_i=\displaystyle\sum_{j=0}^{c-1}\frac{1}{c}\binom{i-1}{c-j-1} q^{i+j-c}p^{c-j},\\ &s_i=\displaystyle\sum_{r=1}^i u_r, \end{array} $$ then $$ \displaystyle\sum_{i=1}^\infty i[s_i^m-(s_i-u_i)^m] $$ converges (I have tested on the computer). But so far, I've not been able to prove it.
The parameters $c$ and $m$ are natural numbers such that $c\geq 2$ and $m\geq 1$.
Can anyone, please, lend me a hand?