Repeating Alternate Divergent Series of Integers

SPECIES

Alternate divergent series of natural numbers repeated twice

$\displaystyle \sum_{n\geq1} (-1)^{n-1}\,\bigg\lfloor\dfrac{n+1}{2}\bigg\rfloor = 1-1+2-2+3-3\pm\cdots = \dfrac{1}{8}$ (unstable)

Erratum: stability flag corrected to unstable by Denis Rogov on 04 May 2023

Alternate divergent series of repeating units

$\displaystyle \sum_{n\geq1} (-1)^{n-1}\,\dfrac{10^n-1}{9} = 1-11+111-1111+11111\pm\cdots = \dfrac{1}{22}$ (stable)

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