There are several ways of computing the partial sum formulas of many summations, but is there a technique that can approximate a closed form for any summation?
So far I found for $\sum_{x=0}^{n} \frac{1}{x}$, Euler suggested an integration technique to compute fractional inputs of a Harmonic partial sum formula. This was used to create the special function $H_{N}$. However, can this or similar technique be applied to approximate all the partial sum formulas of any summation?
If there is an integration technique I would like to see this applied to $\sum_{x=0}^{n}{x\ln(x)}$ and create an imaginary special function, especially for non-integer inputs.