Aperiodic Alternate Divergent Series of Integers

SPECIES

Alternate divergent series of natural numbers (starting from 1)

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

Historical note: Gottfried W. Leibniz (1646–1716) determined the sum of the alternate divergent series of natural numbers in a letter to Christian Wolff dated 1713.

Alternate divergent series of odd numbers (starting from 1)

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

Alternate divergent series of square numbers (starting from 1)

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

Alternate divergent series of cubic numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n^3 = 1-8+27-64+125-216\pm\cdots = -\dfrac{1}{8}$ (stable)

Alternate divergent series of quartic numbers (starting from 1)

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

Alternate divergent series of quintic numbers (starting from 1)

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

Alternate divergent series of triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, \dfrac{n(n+1)}{2} = 1-3+6-10+15-21\pm\cdots = \dfrac{1}{8}$ (stable)

Alternate divergent series of tetrahedral numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, \dfrac{n(n+1)(n+2)}{6} = 1-4+10-20+35-56\pm\cdots = \dfrac{1}{16}$ (stable)

Alternate divergent series of pentachoron numbers (starting from 1)

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, \dfrac{n(n+1)(n+2)(n+3)}{24} = 1-5+15-35\pm\cdots = \dfrac{1}{32}$ (stable)

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 2^n = 1-2+4-8+16-32\pm\cdots = \dfrac{1}{3}$ (stable)

Historical note: Found in 1673 by Gottfried W. Leibniz (1646–1716) by applying Nicholas Mercator’s method. This is also the first known successful attempt to finding out the sum of a divergent series of integers.

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 3^n = 1-3+9-27+81-243\pm\cdots = \dfrac{1}{4}$ (stable)

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 4^n = 1-4+16-64+256-1024\pm\cdots = \dfrac{1}{5}$ (stable)

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

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, 5^n = 1-5+25-125+625-3125\pm\cdots = \dfrac{1}{6}$ (stable)

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

$\displaystyle \sum_{n\geq0} (-1)^n \, n! = 1-1+2-6+24-120\pm\cdots \quad \textit{(stable)} \\ = e \, E_1(1) = G \approx 0.596347362323194074341078499369279376074177860152548\ldots$

$\displaystyle \sum_{n\geq1} (-1)^{n-1} \, n! = 1-2+6-24+120-720\pm\cdots \quad \textit{(stable)} \\ = 1 \,-\, e E_1(1) = 1 \,-\, G \approx 0.403652637676805925658921500630720623925822139\ldots$