Aperiodic Alternate Divergent Series of Integers

VARIETIES

Alternate divergent series of even numbers (starting from 2)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 2n = 2-4+6-8+10-12\pm\cdots = \dfrac{1}{2}$ (stable)

Alternate divergent series of multiples of 3 (starting from 3)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 3n = 3-6+9-12+15-18\pm\cdots = \dfrac{3}{4}$ (stable)

Alternate divergent series of multiples of 4 (starting from 4)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 4n = 4-8+12-16+20-24\pm\cdots = 1$ (stable)

Alternate divergent series of multiples of 5 (starting from 5)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 5n = 5-10+15-20+25-30\pm\cdots = \dfrac{5}{4}$ (stable)

Alternate divergent series of oblong numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n(n+1) = 2-6+12-20+30-42\pm\cdots = \dfrac{1}{4}$ (stable)

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