Aperiodic Alternate Divergent Series of Integers

SECTIONS

Alternate divergent series of polygonal numbers of order r (starting from 1)

\displaystyle \sum_{n\geq1} (-1)^{n-1} \, P_r(n) = P_r(1)-P_r(2)+P_r(3)-P_r(4)\pm\cdots = \dfrac{4-r}{8}   (stable)