Plot probability density functions and probability distribution functions?

Hi guys:

I am plotting a few probability density functions and probability distribution functions these days. I know these functions are far from normal, however, “Plot” function usually cannot give the result in 3 days. All the summation in the equations should be up to infinity. I truncated somewhere as the machine is not able to do that. Also, as pdf and cdf, these functions should be strictly positive, which I used to check up to which number I can safely truncate. Expecting someone who can show me a more efficient way to do draw the graph.

Please see the attachment for details.

Thanks in advance.