Monotonic Divergent Series of Real Numbers

GENERA

$\displaystyle \sum_{n\geq 1} \dfrac{\ln n}{n^r} = -\zeta'(r)$ (semi-stable)

$\displaystyle \sum_{n\geq 1} \dfrac{(\ln n)^2}{n^r} = \zeta''(r)$ (semi-stable)

$\displaystyle \sum_{n\geq 1} \dfrac{(\ln n)^3}{n^r} = -\zeta^{(3)}(r)$ (semi-stable)

$\displaystyle \sum_{n\geq 1} \dfrac{(\ln n)^m}{n^r} = (-1)^m\,\zeta^{(m)}(r)$ (semi-stable)

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