How to Extend the Sum of Any* Function

The antidifference operator Δ^(-1) inverts the forward difference operator Δf(x) = f(x+1) - f(x), mirroring how antiderivatives invert derivatives in continuous calculus.

The general summation extension method unifies previous special cases (harmonic numbers, factorials) under one framework applicable to broad function classes.

Telescoping sums exploit cancellation in summations of differences, providing a discrete analog to the fundamental theorem of calculus.