My goal is to solve a recursive summation without iteration (a lot to ask for I know). Is there a technique for converting a recursive summation into an integral?
For example, these two recursive summations: $$ F_n = \sum_{k=n-1}^{n} F_k $$
$$ L_{x} = \sum_{t=0}^{180} \sum_{k=x+t}^{1000} L_k $$
How can I turn either of these into an integral or a closed form equation? I'm specifically interested in the second (double summation) example.
Better yet, is there a universal solution for turning any arbitrary recursive summation into integral form?