I have a very messy function. It consists sums four levels deep, and the inner-most term is itself quite messy.
$ \sum \sum \sum \sum (\mbox{stuff})$
I can't find a closed form for this function. However, I don't need an exact closed form; I'm only interested in the asymptotic behavior. One idea I have is to approximate the sum using integrals. Would that work? What are some other techniques I can use to upper and/or lower bound a function when the sum is too messy to get into closed form?
Edit: one of the formulas I'm working with looks like this: $\sum_{x_1 = 1}^n \sum_{x_2 = 1}^n \sum_{y_1 = 1}^n \sum_{y_2=1}^n \left( n^{-2} \left(\frac{n- y_2 + y_1 - 1}{n}\right)^{x_2 - x_1 - 1} \right)$