sum command...

[AlexFekken]

Ah, the plot thickens...

The "sophisticated" and "complex" work that you are referring to is simply a re-labelling of the partial sum. It does not do anything significant other than putting a name to the partial sums so that it looks "nicer" when you are using variable bounds.

But your original post isn't about that, it is about calculations with explicit bounds and the Psi function isn't going to help you one bit there, whereas actually doing the sum by going through the loop does (if it doesn't bump into its limitations first).