Aperiodic Monotonic Divergent Series of Integers

SECTIONS

Monotonic divergent series of binomial numbers to the 1st power (starting from 1)

$\displaystyle \sum_{n\geq0} (a+bn) = a + (a+b) + (a+2b) + (a+3b) + \cdots = -\dfrac{a}{2} + \dfrac{5b}{12} \\ \textit{(unstable)}$

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

$\displaystyle \sum_{n\geq1} P_r(n) = P_r(1) + P_r(2) + P_r(3) + P_r(4) + \cdots = \dfrac{r-4}{24} \\ \textit{(unstable)}$

Monotonic convergent series of reciprocals of polygonal numbers of order r (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{1}{P_r(n)} = \dfrac{1}{P_r(1)} + \dfrac{1}{P_r(2)} + \dfrac{1}{P_r(3)} + \dfrac{1}{P_r(4)} + \cdots = \dfrac{r-2-2H_\frac{2}{r-2}}{r-4} \\ \textit{(convergent if } r>2 \textit{)}$

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