Aperiodic Monotonic Divergent Series of Integers

HYPERGENERA

k-shifted monotonic divergent series of natural numbers to the power r

$\displaystyle \sum_{\substack{n\geq 1 \\ k \text{ shifts}}} n^r = \underbrace{0+\cdots+0}_{k\text{ shifts}}+1+2^r+3^r+4^r+\cdots = \zeta(-r) \,+\, \dfrac{(-k)^{r+1}}{r\!+\!1} \quad \textit{(unstable)}$

SUPERGENERA

Monotonic divergent series of natural numbers to the power r (starting from 0)

$\displaystyle \sum_{n\geq0} n^r = 0\,+\,1\,+\,2^r\,+\,3^r\,+\,4^r\,+\,\cdots = \zeta(-r) \,+\, \dfrac{(-1)^{r+1}}{r\!+\!1} \qquad \textit{(unstable)}$

GENERA

Monotonic divergent series of natural numbers to the power r (starting from 1)

$\displaystyle \sum_{n\geq1} n^r = 1\,+\,2^r\,+\,3^r\,+\,4^r\,+\,\cdots = \zeta(-r) = (-1)^r \, \dfrac{B_{r+1}}{r+1} \qquad \textit{(unstable)}$

Monotonic divergent series of simplex numbers of order r (starting from 1)

$\displaystyle \sum_{n\geq1} S_r(n) = S_r(1)\,+\,S_r(2)\,+\,S_r(3)\,+\,S_r(4)\,+\,\cdots = (-1)^{r+1} \, G_{r+1} \qquad \textit{(unstable)}$

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