Aperiodic Monotonic Divergent Series of Integers

SPECIES

Monotonic divergent series of natural numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n = 1+2+3+4+5+6+\cdots = G_2 = -\dfrac{1}{12}$ (unstable)

Monotonic divergent series of odd numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (2n-1) = 1+3+5+7+9+11+\cdots = \dfrac{1}{3}$ (unstable)

Monotonic divergent series of Hilbert numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (4n-3) = 1+5+9+13+17+21+\cdots = \dfrac{7}{6}$ (unstable)

Monotonic divergent series of square numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n^2 = 1+4+9+16+25+36+\cdots = 0$ (unstable)

Monotonic divergent series of cubic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n^3 = 1+8+27+64+125+216+\cdots = \dfrac{1}{120}$ (unstable)

Monotonic divergent series of quartic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n^4 = 1+16+81+256+625+1296+\cdots = 0$ (unstable)

Monotonic divergent series of quintic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n^5 = 1+32+243+1024+3125+\cdots = -\dfrac{1}{252}$ (unstable)

Monotonic divergent series of triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{n(n+1)}{2} = 1+3+6+10+15+21+\cdots = -G_3 = -\dfrac{1}{24}$ (unstable)

Monotonic divergent series of tetrahedral numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{n(n+1)(n+2)}{6} = 1+4+10+20+35+56+\cdots = G_4 = -\dfrac{19}{720}$ (unstable)

Monotonic divergent series of pentachoric numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{n(n+1)(n+2)(n+3)}{24} = 1+5+15+35+70+126+\cdots = -G_5 = -\dfrac{3}{160}$ (unstable)

Monotonic divergent series of pentagonal numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{n(3n-1)}{2} = 1+5+12+22+35+51+\cdots = \dfrac{1}{24}$ (unstable)

Monotonic divergent series of hexagonal numbers (starting from 1)

$\displaystyle \sum_{n\geq1} n(2n-1) = 1+6+15+28+45+66+\cdots = \dfrac{1}{12}$ (unstable)

Monotonic divergent series of centered square numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (2n^2-2n+1) = 1+5+13+25+41+61+\cdots = -\dfrac{1}{3}$ (unstable)

Monotonic divergent series of square-triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{(17+12\sqrt{2})^n + (17-12\sqrt{2})^n - 2}{32}$

$= 1+36+1225+41616+1413721+48024900+\cdots = 0$ (unstable)

Monotonic divergent series of powers of 2 (starting from 1)

$\displaystyle \sum_{n\geq0} 2^n = 1+2+4+8+16+32+\cdots = -1$ (stable)

Monotonic divergent series of powers of 3 (starting from 1)

$\displaystyle \sum_{n\geq0} 3^n = 1+3+9+27+81+243+\cdots = -\dfrac{1}{2}$ (stable)

Monotonic divergent series of powers of 4 (starting from 1)

$\displaystyle \sum_{n\geq0} 4^n = 1+4+16+64+256+1024+\cdots = -\dfrac{1}{3}$ (stable)

Monotonic divergent series of powers of 5 (starting from 1)

$\displaystyle \sum_{n\geq0} 5^n = 1+5+25+125+625+3125+\cdots = -\dfrac{1}{4}$ (stable)

Monotonic divergent series of Mersenne numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (2^n-1) = 1+3+7+15+31+63+\cdots = -\dfrac{3}{2}$ (unstable)

Monotonic divergent series of natural numbers times powers of 2 (starting from 1)

$\displaystyle \sum_{n\geq1} n2^{n-1} = 1+4+12+32+80+192+\cdots = 1$ (stable)

Monotonic divergent series of factorial numbers (starting from 0 or 1)

$\displaystyle \sum_{n\geq0} n! = 1+1+2+6+24+120+\cdots = \dfrac{\text{Ei}(1)}{e} = \dfrac{\text{li}(e)}{e} \quad \textit{(stable)} \\ \approx 0.697174883235066068765478681919551595317175430954369517320054807\ldots$

$\displaystyle \sum_{n\geq1} n! = 1+2+6+24+120+\cdots = \dfrac{\text{Ei}(1)}{e} \,-\,1 = \dfrac{\text{li}(e)}{e} \,-\, 1 \quad \textit{(stable)} \\ \approx -0.30282511676493393123452131808044840468282456904563048267994519\ldots$