Periodic Alternate Divergent Series of Integers

SPECIES

Periodic alternate Grandi series [1–1]

$\displaystyle \sum_{n\geq 1} (-1)^{n-1} = 1-1+1-1+1-1\pm\cdots = \dfrac{1}{2}$ (stable)

Historical note: Even though named after Luigi Guido Grandi (1671–1742) who determined this sum in 1703 in his book “Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita” (L.G. Grandi, 1703) using a binomial expansion, the sum of the alternate divergent series was actually determined first in 1674 by Gottfried W. Leibniz (1646–1716) in his essay “De Triangulo Harmonico” (G.W. Leibniz, 1674). This is also the second known successful attempt to finding out the sum of a divergent series of integers.
Jacob Bernoulli (1654 –1705) also dealt with this series in the third part of his “Positiones arithmeticae de seriebus infinitis” (J. Bernoulli, 1696).
Leonhard Euler (1707–1783) also determined the sum of this series in his paper “De seriebus divergentibus” (L. Euler, 1746) read to the Berlin Academy in 1754, but published only in 1760, using his finite forward difference method.

Periodic alternate Lagrange series [1+0–1]

$\displaystyle \sum_{n\geq 1} \left(2+3\bigg\lfloor\dfrac{n-1}{3}\bigg\rfloor-n\right) = 1+0-1+1+0-1+1+0-1\pm\cdots = \dfrac{2}{3}$ (stable)

Historical note: In an unpublished memorandum sent in the 1780s by Jean-François Callet (1744–1798) to Joseph-Louis Lagrange (1736-1813) for review, the sum of the periodic alternate divergent series [1+0-1] was correctly determined to be 2/3 by J-L. Lagrange. This is the first known explicit determination of the sum of a divergent series of integers with spacing zeros.
However, Daniel Bernoulli (1700–1782) noticed in his essay “De summationibus serierum quarunduam incongrue veris earumque interpretatione atque usu” (D. Bernoulli, 1771) that by inserting zeros into the Grandi series in the right places, he could achieve any value between 0 and 1 as the sum of the modified divergent series.

Periodic alternate divergent series [1–2]

$\displaystyle \sum_{n\geq 1} \dfrac{-1-3(-1)^n}{2} = 1-2+1-2+1-2\pm\cdots = 1$ (unstable)

Periodic alternate divergent series [2–1]

$\displaystyle \sum_{n\geq 1} \dfrac{1-3(-1)^n}{2} = 2-1+2-1+2-1\pm\cdots = \dfrac{1}{2}$ (unstable)

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