Periodic Monotonic Characteristic Series

VARIETIES

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

$\displaystyle \sum_{n\geq1} \chi\left(\bigg\lfloor\frac{n+1}{2}\bigg\rfloor\right) = \displaystyle \sum_{n\geq1} 2 = -1$ (unstable)

$2+2+2+2+2+2+2+2+\cdots = -1$ (unstable)

Periodic monotonic characteristic series of natural numbers repeated 3 times (starting from 1)

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

$3+3+3+3+3+3+3+3+\cdots = -\dfrac{3}{2}$ (unstable)

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