Aperiodic Alternate Divergent Series of Integers

GENERA

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n^r = 1-2^r+3^r-4^r+5^r-6^r\pm\cdots = (2^{r+1}-1) \, \dfrac{B_{r+1}}{r+1}$ (stable)

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, S_r(n) = S_r(1)-S_r(2)+S_r(3)-S_r(4)\pm\cdots = \dfrac{1}{2^{r+1}}$ (stable)

Pages: 1 2 3 4 5 6 7