Find the following sum:
where $x$ is a real number. This is a power series in $x$. In particular, I'm interested in the case $x>0$.
Disclaimer: This is not a homework exercise, I do not know if a closed form solution exists. If it doesn't, exist, then an approximation in terms of well-known functions (not the all-mighty general hypergeometric $_pF_q$, something simpler please) would be desired.