While working on a model of the activity in a slot machines lounge, I encountered that the following formula should be finite:
$$ \displaystyle\sum_{i=1}^\infty\frac{i}{c}\displaystyle\sum_{j=0} ^{c-1}\binom{i-1}{c-j-1}q^{i+j-c}p^{c-j}, $$
but I've not been able to prove it. I would thank very much any suggestion.
Let $S$ denote the sum. Substituting $k = c - 1 - j$ and interchanging the order of summation,
\begin{align*} S &= \frac{1}{c} \sum_{k=0}^{c-1} \frac{p^{k+1}}{k!} \sum_{i=1}^{\infty} \frac{i!}{(i-1-k)!} q^{i-1-k} \\ &= \frac{1}{c} \sum_{k=0}^{c-1} \frac{p^{k+1}}{k!} \left( \frac{\partial}{\partial q}\right)^{k+1} \sum_{i=0}^{\infty} q^{i} \\ &= \frac{1}{c} \sum_{k=0}^{c-1} \frac{p^{k+1}}{k!} \frac{(k+1)!}{(1-q)^{k+2}} \\ &= \frac{1}{c} \sum_{k=0}^{c-1} \frac{k+1}{p} \\ &= \frac{c+1}{2p}. \end{align*}