Periodic Monotonic Divergent Series of Integers

SPECIES

Periodic monotonic series of 0

$\displaystyle \sum_{n\geq m} 0 = 0+0+0+0+0+0+\cdots = 0 \qquad (m \in \mathbb{Z})$ (convergent)

Periodic monotonic divergent series of 1

$\displaystyle \sum_{n\geq m} 1 = 1+1+1+1+1+1+\cdots = -\dfrac{1}{2} \qquad (m \in \mathbb{Z})$ (unstable)

Periodic monotonic divergent series [1+2]

$\displaystyle \sum_{n\geq1} \dfrac{3+(-1)^n}{2} = 1+2+1+2+1+2+\cdots = -1$ (unstable)

Periodic monotonic divergent series [2+1]

$\displaystyle \sum_{n\geq1} \dfrac{3-(-1)^n}{2} = 2+1+2+1+2+1+\cdots = -\dfrac{1}{2}$ (unstable)

Periodic monotonic divergent series [1+2+3]

$\displaystyle \sum_{n\geq1} \left(n+3\left(1-\bigg\lfloor\dfrac{n+2}{3}\bigg\rfloor\right)\right) = 1+2+3+1+2+3+\cdots = -\dfrac{5}{3}$ (unstable)

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