Sometimes we have the
right intuitions, and we even publish those, as Mark Dodds has done on https://www.cantorsparadise.com/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793. He then used expressions such as 'crazy claims' to
refer to some of the findings of Ramanujan, and Grandi.
As the author of Infinis
(http://www.protosociology.de/Download/Pinheiro-Infinis.pdf), I would like to say that infinity is not a place: it
is the absence of one. Things being such, we cannot call it a name, and use
this name to make calculations. Mathematicians know: we indicate infinity, but
we don't use it in mathematical operations.
If you have a false
premise, there is a good chance you will reach a false conclusion. It is not
guaranteed to be the case, since, for instance, if you assume that 0 is equal
to 1/1, and you then swap / for -, you get 0=1/1=1-1=0, or 0=0, therefore a
true conclusion.
An implication that has a
false antecedent will always be true in Mathematics, as we see on http://article.sapub.org/10.5923.j.ap.20170101.03.html, so that, in mathematical proofs (taking away the proof
by contradiction), we always have a true antecedent instead of a false one. A
true antecedent with a false consequent gives us a false implication. A true
antecedent with a true consequent gives us a true implication. Now it makes
sense, since we can tell wrong from right… .
Now, the partial sums in
(1-1)+(1-1)+(1-1)...diverge, since S1=1, S2=0, S3=1, S4=0, and so on. The
partial sums in 1+2+3+4... also diverge, since S1=1, S2=3, S3=6, S4=10, and so
on, so that we know there is no 'place of arrival' for these sums, and
therefore we can never equate them to a certain place, so say A, B or C. There
should be no equal sign there! If we put an equal sign after the sums
(1-1)+(1-1)+(1-1)+…, and 1+2+3+4+…, that is a false assumption, a false
antecedent. We can only have true ones (it is not a proof by
contradiction).
Since we assumed those
sums had a 'place of arrival', we got to true or false conclusions, and to true
implications, what made us have the sensation that we had a proof, but it was
actually just a hoax.
If we deal with the sums
on their own, without adding an equal sign to them, and a 'place of arrival' or
variable, we get to the right conclusions, which are that they both go to
nowhere in particular.
Notice that, in reality,
in Mathematics, we can put an equal sign there, but we then call the result sum
or limit of an expression when n goes
to infinity, what will make the conclusions of Ramanujan, and Grandi impossible
because we then have ∑(-1)n=1-1+1-1+… therefore 1-∑(-1)n=1-(1-1+1-1+…)
therefore 1-∑(-1)n=1-1+1-1+… therefore 1-∑(-1)n=∑(-1)n
therefore ∑(-1)n+∑(-1)n=1, but ∑(-1)n diverges,
and therefore we cannot find a result for ∑(-1)n+∑(-1)n, as we see
on https://math.stackexchange.com/questions/59512/sum-of-infinite-divergent-series, and therefore ∑(-1)n+∑(-1)n is
not 2∑(-1)n, and therefore the ‘crazy’ result does not eventuate.
