Aperiodic Alternate Divergent Series of Integers

SUPERSPECIES

Single-shifted alternate divergent series of natural numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, n = 0+1-2+3-4+5-6\pm\cdots = \dfrac{1}{4}$ (stable)

Single-shifted alternate divergent series of odd numbers (starting from 1)

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

Single-shifted alternate divergent series of square numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, n^2 = 0+1-4+9-16+25-36\pm\cdots = 0$ (stable)

Single-shifted alternate divergent series of cubic numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, n^3 = 0+1-8+27-64+125-216\pm\cdots = -\dfrac{1}{8}$ (stable)

Single-shifted alternate divergent series of quartic numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, n^4 = 0+1-16+81-256+625-1296\pm\cdots = 0$ (stable)

Single-shifted alternate divergent series of quintic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n^5 = 0+1-32+243-1024+3125-7776\pm\cdots = \dfrac{1}{4}$ (stable)

Single-shifted alternate divergent series of triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, \dfrac{n(n+1)}{2} = 0+1-3+6-10+15-21\pm\cdots = \dfrac{1}{8}$ (stable)

Single-shifted alternate divergent series of tetrahedral numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, \dfrac{n(n+1)(n+2)}{6} = 0+1-4+10-20+35\pm\cdots = \dfrac{1}{16}$ (stable)

Single-shifted alternate divergent series of pentachoron numbers (starting from 1)

$\displaystyle \sum_{n\geq0} (-1)^{n-1} \, \dfrac{n(n+1)(n+2)(n+3)}{24} = 0+1-5+15\pm\cdots = \dfrac{1}{32}$ (stable)

Single-shifted alternate divergent series of powers of 2 (starting from 1)

$\displaystyle \sum_{\substack{n\geq0 \\ 1 \text{ shift}}} (-1)^{n-1} \, 2^n = 0+1-2+4-8+16-32\pm\cdots = \dfrac{1}{3}$ (stable)

Single-shifted alternate divergent series of powers of 3 (starting from 1)

$\displaystyle \sum_{\substack{n\geq0 \\ 1 \text{ shift}}} (-1)^{n-1} \, 3^n = 0+1-3+9-27+81-243\pm\cdots = \dfrac{1}{4}$ (stable)

Single-shifted alternate divergent series of powers of 4 (starting from 1)

$\displaystyle \sum_{\substack{n\geq0 \\ 1 \text{ shift}}} (-1)^{n-1} \, 4^n = 0+1-4+16-64+256-1024\pm\cdots = \dfrac{1}{5}$ (stable)

Single-shifted alternate divergent series of powers of 5 (starting from 1)

$\displaystyle \sum_{\substack{n\geq0 \\ 1 \text{ shift}}} (-1)^{n-1} \, 5^n = 0+1-5+25-125+625-3125\pm\cdots = \dfrac{1}{6}$ (stable)

Single-shifted alternate divergent series of Fibonacci numbers (starting from 1–1)

$\displaystyle \sum_{\substack{n\geq1 \\ 1 \text{ shift}}} (-1)^{n-1} \, F_n = 0+1-1+2-3+5-8+13-21\pm\cdots = 1$ (stable)

Single-shifted alternate divergent series of Lucas numbers (starting from 1–3)

$\displaystyle \sum_{\substack{n\geq1 \\ 1 \text{ shift}}} (-1)^{n-1} \, L_n = 0+1-3+4-7+11-18+29-47\pm\cdots = -1$ (stable)

Single-shifted alternate divergent series of Catalan numbers

$\displaystyle \sum_{\substack{n\geq1 \\ 1 \text{ shift}}} (-1)^{n-1} \, C_n = 0+1-2+5-14+42-132+429\pm\cdots = \dfrac{3}{2}$ (stable)

Single-spaced alternate divergent series of natural numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \dfrac{n+1}{2}$

$= 1+0-2+0+3+0-4+0+5+0-6+0\pm\cdots = \dfrac{1}{4}$ (stable)

Single-spaced alternate divergent series of odd numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, n$

$= 1+0-3+0+5+0-7+0+9+0-11+0\pm\cdots = 0$ (stable)

Single-spaced alternate divergent series of square numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \left(\dfrac{n+1}{2}\right)^2$

$= 1+0-4+0+9+0-16+0+25+0-36+0\pm\cdots = 0$ (stable)

Single-spaced alternate divergent series of cubic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \left(\dfrac{n+1}{2}\right)^3$

$= 1+0-8+0+27+0-64+0+125+0-216+0\pm\cdots = -\dfrac{1}{8}$ (stable)

Single-spaced alternate divergent series of quartic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \left(\dfrac{n+1}{2}\right)^4$

$= 1+0-16+0+81+0-256+0+625+0-1296+0\pm\cdots = 0$ (stable)

Single-spaced alternate divergent series of quintic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \left(\dfrac{n+1}{2}\right)^5$

$= 1+0-32+0+243+0-1024+0+3125+0-7776+0\pm\cdots = \dfrac{1}{4}$ (stable)

Single-spaced alternate divergent series of triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \dfrac{(n+1)(n+3)}{8}$

$= 1+0-3+0+6+0-10+0+15+0-21+0\pm\cdots = \dfrac{1}{8}$ (stable)

Single-spaced alternate divergent series of tetrahedral numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \dfrac{(n+1)(n+3)(n+5)}{48}$

$= 1+0-4+0+10+0-20+0+35+0-56+0\pm\cdots = \dfrac{1}{16}$ (stable)

Single-spaced alternate divergent series of pentachoron numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, \dfrac{(n+1)(n+3)(n+5)(n+7)}{384}$

$= 1+0-5+0+15+0-35+0+70+0-126+0\pm\cdots = \dfrac{1}{32}$ (stable)

Single-spaced alternate divergent series of powers of 2 (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} \, (-2)^\frac{n-1}{2}$

$= 1+0-2+0+4+0-8+0+16+0-32+0\pm\cdots = \dfrac{1}{3}$ (stable)

Single-spaced alternate divergent series of powers of 3 (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} \, (-3)^\frac{n-1}{2}$

$= 1+0-3+0+9+0-27+0+81+0-243+0\pm\cdots = \dfrac{1}{4}$ (stable)

Single-spaced alternate divergent series of powers of 4 (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} \, (-4)^\frac{n-1}{2}$

$= 1+0-4+0+16+0-64+0+256+0-1024+0\pm\cdots = \dfrac{1}{5}$ (stable)

Single-spaced alternate divergent series of powers of 5 (starting from 1)

$\displaystyle \sum_{n\geq1} \dfrac{1-(-1)^n}{2} \, (-5)^\frac{n-1}{2}$

$= 1+0-5+0+25+0-125+0+625+0-3125+0\pm\cdots = \dfrac{1}{6}$ (stable)

Single-spaced alternate divergent series of Fibonacci numbers (starting from 1–1)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, F_{\lfloor\frac{n+1}{2}\rfloor}$

$= 1+0-1+0+2+0-3+0+5+0-8+0+13+0\pm\cdots = 1$ (stable)

Single-spaced alternate divergent series of Lucas numbers (starting from 1–3)

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, L_{\lfloor\frac{n+1}{2}\rfloor}$

$= 1+0-3+0+4+0-7+0+11+0-18+0+29+0\pm\cdots = -1$ (stable)

Single-spaced alternate divergent series of Catalan numbers

$\displaystyle \sum_{n\geq1} (-1)^\frac{n-1}{2} \, \dfrac{1-(-1)^n}{2} \, C_{\lfloor\frac{n+1}{2}\rfloor}$

$= 1+0-2+0+5+0-14+0+42+0-132+0\pm\cdots = \dfrac{3}{2}$ (stable)

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