Repeating Monotonic Divergent Series of Integers

SPECIES

Monotonic divergent series of natural numbers repeated twice (starting from 1)

$\displaystyle \sum_{n\geq1} \bigg\lfloor\dfrac{n+1}{2}\bigg\rfloor = 1+1+2+2+3+3+\cdots = -\dfrac{1}{24}$ (unstable)

Monotonic divergent series of natural numbers repeated 3 times (starting from 1)

$\displaystyle \sum_{n\geq1} \bigg\lfloor\dfrac{n+2}{3}\bigg\rfloor = 1+1+1+2+2+2+3+3+3+\cdots = \dfrac{1}{36}$ (unstable)

Monotonic divergent series of natural numbers repeated 2ⁿ times (starting from 1)

$\displaystyle \sum_{n\geq1} \lfloor\log_2(n+1)\rfloor = 1+1+2+2+2+2+3+\cdots = 2$ (unstable)

Monotonic divergent series of natural numbers repeated 2^n-1 times (starting from 1)

$\displaystyle \sum_{n\geq1} \left(\lfloor\log_2(n)\rfloor+1\right) = 1+2+2+3+3+3+3+4+\cdots = \dfrac{3}{2}$ (unstable)

Monotonic divergent series of repeated units

$\displaystyle \sum_{n\geq1} \dfrac{10^n-1}{9} = 1+11+111+1111+11111+\cdots = -\dfrac{11}{162}$ (unstable)

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