There is no question that mathematics is, after a certain level, just a game with symbols that an elite priesthood jack off to. The financial elite have their money; the mathematical elite, symbolic structures. Hence, while it is true that the series 1+2+3+… converges in the standard sense, mathematicians have found it amusing, and useful for shonky physicists working in areas dripping in infinities, such as string theory and quantum field theory, to come up with a useful result.

Thus, the series 1-1+1-1+…, diverges in the standard sense, but a Cesaro summation can be given, defined as the limit of the arithmetic mean of the partial sums of the series. Then 1-1+1-1+…=1/2, the value which the series oscillates around. See: G. H. Hardy, “Divergent Series,” (Clarendon Press, Oxford, 1949).

But, not all series can be given a Cesaro sum, such as 1+2+3+…, and that is where the Euler/Riemann zeta function comes in. Zeta function regulation involves defining the Riemann zeta function for all values of z as the analytic continuum, which means extending the definition of the function from reals to functions of complex numbers. Thus:

ξ(x)= S(x)= 1+ 1/2x + 1/3x =…

If x= -1 is put into this function, then one gets ξ(-1)= -1/12.

Some may conclude that 1+2+3+…, therefore equals -1/12. This, however is incorrect, because strictly speaking divergent series in the standard sense can be assigned a number of different values based on analytic continuum, so the “=” is not used in a standard sense at all. For example, the series 1-1+1-1+…, could be assigned the value 1/3. One can obtain some truly bizarre results such as:

1+1+1+…= -1/2

1+4+9+… =0

to name some classics. Terry Tao discusses these in his paper, “The Euler-Maclaurin Formula, Bernoulli Numbers, the Zeta Function, and Real-Variable Analytic Continuation,” April 10, 2010, at https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/. Tao sees a problem here: “Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous. But as it stands, the formulae look “wrong” for several reasons. Most obviously, the summands on the left are all positive, but the right-hand sides can be zero or negative.” A “problem,” indeed!

Even given the zeta function result, if “=” is given its standard meaning then we have a contradictory result, precisely because the summands on the left are positive and non-zero, but the right-hand side is zero or negative, which gives a classical contradiction p&~p. But, if “=” is given a non-standard meaning, then the physicists can’t use this reasoning in their shonky physics, to avoid infinites.

It is known, but often ignored, that with bracket manipulation within an infinite series, and with the rearrangement of terms, inconsistencies can be generated, even for convergent series, so that one can get a sum, which is any real number, with a bit of ingenuity. Thus, although 1-1+1-1+…= ½, via the analytic continuation of the zeta function, if one adds brackets, one can get;

(1-1)+(1-1)+(1-1)+…=0+0+0+0=…, which is not 1/2. My guess is it is 0, or should be 0.

With bracket rearrangement, one can “prove” that 1-1+1-1…=1/2, even without use of the zeta function. Let S=1-1+1-1+…, then:

1- S= 1-(1-1+1-1+1-…) = 1-1+1-1+… =S. So, 2S=1, or S=1/2.

Consider again, S = 1 + 2 + 3 + 4 + 5 + · · · . We have −3S= (1 − 4)S = (1 + 2 + 3 + 4 + 5 + · · ·) − 2(2 + 4 + 6 + · · ·) = 1 − 2 + 3 − 4 + 5 − · · · = 1 − (2 − 3 + 4 − 5 + · · ·) = 1 − (1 − 2 + 3 − 4 + 5 − · · ·) − (1 − 1 + 1 − 1 + · · ·) = 1 + 3S − 1/2, and we conclude that −6s = 1/2, that is 1 + 2 + 3 + 4 + 5 + · · · = − 1/12 .

Here is another so-called “proof”:

Term-by-term summation used in Numberphile's video

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + … = ?

S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2

S2 = 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …

2S2 = 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + …

+ 1 − 2 + 3 − 4 + 5 − 6 + 7 + …

= 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2

S2 = 1/4

S − S2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + …

− 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 + …

= 0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + … = 4S

S – 1/4 = 4S ⇒ S = – 1/12

But as a counter to Numberphile, consider this:

1+2+3+4+… = 1+(1+1)+(1+2)+(1+3)+… = (1+1+1+1+…) + (1+2+3+4+…).

Therefore -1/2 + -1/12 = -1/12, which implies that -1/2 =0, which implies that 1=0, an absolute inconsistency.

Clearly, then, all of the above results are rubbish and thus should be a caution about regarding infinite series as things which can be manipulated according to the mathematical rules normally applied to finite sums. However, noting this leads to some even more interesting results when we re-examine the issue about .999… being equal to 1, in my next arousing paper.

I've found the venue:

https://en.wikipedia.org/wiki/Kabbalah_Centre