I would like to solve a long summation related to mean residual life for a new discrete distribution but I have difficulty to solve the following summation (a part of larger summation), which is given as $$ \sum_{j=x}^{\infty} p^j \Phi(p,1,j+2), $$
where $0<p<1$ is unknown and the $x=0,1,2,...$.
How to solve the above summation?
The answer I think should be in terms of $x$ (probably in the form of $p^x$) and this is difficult for me to solve. Any hint or solution or reference to papers are welcomed.
