The divergent series 1+1+…
In the present article I intend give plausible value for the divergent series 1+1+… according with Euler’s results and with relaxed properties of associativity, distributivity and so on.Most of the following results are rigorously nonsense, ” a facon de ecrire”
For example, we know that so
(sum by columns…)
and so that is s=0 so that zero is a plausibile value for s (actually it’s the most accreditated!).
plausible value for s
, n positive integer , is plausible too:
is plausibile too, and the same for
plausible
, n positive integer, is plausible too:
that is is plausible too. And similarly for n integer positive.
plausible
plausible too:
is plausible too, and the same for all n.
Therefore starting with the initial value 0 we know that all negative rational numbers are plausible.
Let’s recall for example the value :
starting from the well known fact we have:
.
But we haven’t only rational negative numbers:
(sum the numbers in column !)
and then
is plausible too: a complex imaginary number!
and solving
are plausible values too.
This calculation gives also one continued fraction for s:
References:
Leonard Euler: Institutiones calculi differentialis
G.H.Hardy: Divergent Series, At the Clarendon Press 1949
Oddifreddi: https://www.youtube.com/watch?v=FCSaXPYJ1zk
https://mathoverflow.net/questions/85678/eulers-divergent-series-sum-n-1n-what-is-known-about-the-resulting-consta/85683
Wikipedia https://it.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7
Egregio estensore,
Quello sulle serie è un risultato sconcertante, che mette in crisi le certezze di ogni matematico formatosi nel clima neoscolastico di molte delle nostre attuali università. Bisogna tornare a lavorare sulle opere dei grandi del passato per capire fino a che punto si è persa la cognizione della materia in favore delle fumisterie della filosofia analitica e dei formalisti. In questo caso si tratta di un risultato riconducibile a Eulero, di cui già Lagrangia (meglio noto nella forma gallicizzata Lagrange) diceva:
«Lisez Euler, lisez Euler, c’est notre maître à tous».