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)