Periodic Monotonic Characteristic Series

SPECIES

Periodic monotonic characteristic series of natural numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi(n) = \displaystyle \sum_{n\geq1} 1 = -\dfrac{1}{2}$ (unstable)

$1+1+1+1+1+1+1+1+\cdots = -\dfrac{1}{2}$ (unstable)

Periodic monotonic characteristic series of even numbers (starting from 2)

$\displaystyle \sum_{n\geq1} \chi(2n) = \displaystyle \sum_{n\geq1} \dfrac{1+(-1)^n}{2} = -\dfrac{1}{2}$ (unstable)

$0+1+0+1+0+1+0+1+\cdots = -\dfrac{1}{2}$ (unstable)

Periodic monotonic characteristic series of odd numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi(2n-1) = \displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} = 0$ (unstable)

$1+0+1+0+1+0+1+0+\cdots = 0$ (unstable)

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