1/2 = (1 + 0)/2 and it is used to smooth out Gibbs thingies at points of discontinuity/incompatibility.The theory of dynamical systems (Lotka-Voltera model, strange attractors, etc.) and practically all modern theories in economy, sociology or biology would not exist if Cauchy's descendants (Poincaré?) dropped the problem at that point - vide limit sets. (I agree that the S=1/2 makes no sense, though.)the [$]n[$]th-term test for divergence:

If [$]\lim_{n\to\infty}a_{n}\ne 0[$] or if the limit does not exist, then [$]\sum _{n=1}^{\infty }a_{n}[$] diverges

Which is what we've got here

If S = 1/2 is not to your liking you can have

S = 0 = (1-1) + (1-1)+...

or

S = 1 = 1 + (-1+1) + (-1+1)+ ...