Periodic Alternate Divergent Series of Integers

SUPERSPECIES

Single-spaced Grandi series [1–1]

$\displaystyle \sum_{n\geq1} (-1)^{\lfloor\frac{n-1}{2}\rfloor} \, \dfrac{1-(-1)^n}{2} = 1+0-1+0+1+0-1+0\pm\cdots = \dfrac{1}{2}$ (stable)

Single-spaced and single-shifted Grandi series [1–1]

$\displaystyle \sum_{n\geq1} (-1)^{\lfloor\frac{n-1}{2}\rfloor} \, \dfrac{1+(-1)^n}{2} = 0+1+0-1+0+1+0-1+0\pm\cdots = \dfrac{1}{2}$ (stable)

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