Aperiodic Monotonic Divergent Series of Integers

SUPERSPECIES

Single-shifted monotonic divergent series of natural numbers

$\displaystyle \sum_{n\geq0} n = 0+1+2+3+4+5+6+\cdots = \dfrac{5}{12}$ (unstable)

Single-shifted monotonic divergent series of odd numbers

$\displaystyle \sum_{\substack{n\geq1 \\ 1 \text{ shift}}} (2n-1) = 0+1+3+5+7+9+11+\cdots = \dfrac{7}{3}$ (unstable)

Single-shifted monotonic divergent series of square numbers

$\displaystyle \sum_{n\geq0} n^2 = 0+1+4+9+16+25+36+\cdots = -\dfrac{1}{3}$ (unstable)

Single-shifted monotonic divergent series of cubic numbers

$\displaystyle \sum_{n\geq0} n^3 = 0+1+8+27+64+125+216+\cdots = \dfrac{31}{120}$ (unstable)

Single-shifted monotonic divergent series of quartic numbers

$\displaystyle \sum_{n\geq0} n^4 = 0+1+16+81+256+625+1296+\cdots = -\dfrac{1}{5}$ (unstable)

Single-shifted monotonic divergent series of quintic numbers

$\displaystyle \sum_{n\geq0} n^5 = 0+1+32+243+1024+3125+7776+\cdots = \dfrac{41}{252}$ (unstable)

Single-spaced monotonic divergent series of natural numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n+1}{2}\right)\,\dfrac{1-(-1)^n}{2} = 1+0+2+0+3+0+4+0+\cdots = \dfrac{1}{24}$ (unstable)

Single-spaced monotonic divergent series of odd numbers

$\displaystyle \sum_{n\geq1} n\,\dfrac{1-(-1)^n}{2} = 1+0+3+0+5+0+7+0+\cdots = \dfrac{1}{12}$ (unstable)

Single-spaced monotonic divergent series of square numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n+1}{2}\right)^2\,\dfrac{1-(-1)^n}{2} = 1+0+4+0+9+0+16+0+\cdots = \dfrac{1}{24}$ (unstable)

Single-spaced monotonic divergent series of cubic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n+1}{2}\right)^3\,\dfrac{1-(-1)^n}{2} = 1+0+8+0+27+0+\cdots = \dfrac{23}{960}$ (unstable)

Single-spaced monotonic divergent series of quartic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n+1}{2}\right)^4\,\dfrac{1-(-1)^n}{2} = 1+0+16+0+81+0+\cdots = \dfrac{1}{160}$ (unstable)

Erratum: above sum corrected by Denis Rogov on 28 Sep 2023

Single-spaced monotonic divergent series of quintic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n+1}{2}\right)^5\,\dfrac{1-(-1)^n}{2} = 1+0+32+0+243+0+\cdots = -\dfrac{11}{8064}$ (unstable)

Erratum: above sum corrected by Denis Rogov on 30 Sep 2023

Single-spaced and single-shifted monotonic divergent series of natural numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n}{2}\right)\,\dfrac{1+(-1)^n}{2} = 0+1+0+2+0+3+0+4+0+\cdots = -\dfrac{1}{12}$ (unstable)

Single-spaced and single-shifted monotonic divergent series of odd numbers

$\displaystyle \sum_{n\geq1} (n-1)\,\dfrac{1+(-1)^n}{2} = 0+1+0+3+0+5+0+7+0+\cdots = \dfrac{1}{3}$ (unstable)

Single-spaced and single-shifted monotonic divergent series of square numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n}{2}\right)^2\,\dfrac{1+(-1)^n}{2} = 0+1+0+4+0+9+0+16+0+\cdots = 0$ (unstable)

Single-spaced and single-shifted monotonic divergent series of cubic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n}{2}\right)^3\,\dfrac{1+(-1)^n}{2} = 0+1+0+8+0+27+0+\cdots = \dfrac{1}{120}$ (unstable)

Single-spaced and single-shifted monotonic divergent series of quartic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n}{2}\right)^4\,\dfrac{1+(-1)^n}{2} = 0+1+0+16+0+81+0+\cdots = 0$ (unstable)

Single-spaced and single-shifted monotonic divergent series of quintic numbers

$\displaystyle \sum_{n\geq1} \left(\dfrac{n}{2}\right)^5\,\dfrac{1+(-1)^n}{2} = 0+1+0+32+0+243+0+\cdots = -\dfrac{1}{252}$ (unstable)

Pages: 1 2 3 4 5 6 7 8