Why -1/12 is a gold nugget

What people said

Strong rec orion · 2016-01 · Mathematics Stack Exchange · on “Do these infinite series converge to a finite limit?”

These are regularized sums. The first one is actually the one that you already wrote in the first line (adding zero doesn't do anything). The second turns out to regularize to zero. Read up on zeta regularization and watch these excellent videos [LINK] [this video] And you can generalize $\sum_{n=0}^\infty n^k = \zeta(-k)$ where $\zeta$ is the Riemann zeta function - you can see from the definition of this function, that all we do is to put in a number outside the convergence domain of original series, so the result is not a true sum of the series (which diverges) but analytic continuation that extends the domain beyond what the series can do.

Recommends jimjim · 2015-05 · Mathematics Stack Exchange · on “Can there be a sequence which is not convergent but has accumulation points?”

Look at cesaro summibility, you get few ideas by knowing that convergence in the sense of your book is only the classical convergence. If one considers partial sums as a sequence then you would see that unbounded sequences can still converge in a sense e.g. see this vide [LINK] and this [this video]

Recommends Gabriel Bello · 2016-10 · Mathematics Stack Exchange · on “Could $4+2+4+2+4+2+\cdots = -1 $?”

You have to be very careful when claiming that infinite, divergent sums add to a real number. Strictly speaking, this makes no sense since you cannot manipulate indefinite sums the way you please. The video @Jonas12 posted is useful, but I would also recommend watching: [this video] to further comprehend the nature of this result.

Recommends Vincenzo Oliva · 2018-01 · Mathematics Stack Exchange · on “What does $1+2+3+...=-\frac{1}{12}$ mean?”

Or, your brain is right and that Numberphile video is garbage. It's not a true equality. The LHS and RHS are inherently connected, but not equal in the usual sense. That divergent sum can be *regularised* to that value (unique, if we abide by certain rules). By the way, take a look at this other video [this video] from Numberphile, it's a sort of amendment.

Recommends George · 2020-01 · Mathematics Stack Exchange · on “'Limits' of divergent series'”

I was watching a Numberphile video [this video] 'Why $-\frac{1}{12}$ is a gold nugget', in which Professor Edward Frenkel was discussing assigning values to divergent series'. He mentioned that Euler, being the mathematical gangster he was, used to manipulate infinite series in attempt to assign a value to them. For example: $$1+2+3+4+...=-\frac{1}{12},$$ $$1^2+2^2+3^2+4^2+...=0,$$ $$1^3+2^3+3^3+4^3+...=-\frac{1}{120}.$$ Question 1 He mentions that around 100 years later Riemann proved Euler's suggestions with the Zeta function. I was just wondering what Euler's original method was in assigning these values? Question 2 He also later mentions there is a now, mathematically rigorous framework in wh

Cites pshmath0 · 2014-04 · Mathematics Stack Exchange · on “Invalid use of the analytic continuation of the Riemann zeta function?”

Watching this video [this video] on You Tube I got the impression that some sciences (in this case physics) use the analytic continuation of the Riemann zeta function without justification. Maybe this is just my interpretation of what was being said, but I will continue: For example, on logically arriving at something like $$\sum_{n=1}^\infty n^2,$$ one might naively replace this with $\zeta(-2)$, which has value $0$ under the analytic continuation of the Riemann zeta function. My question is simply: do some sciences really do this? Surely any results obtained this way are suspect until proven by alternative methods? In classical mathematics I've read that Euler used divergent definite integrals

Cites ziggurism · 2016-05 · Mathematics Stack Exchange · on “Can we show that $1+2+3+\dotsb=-\frac{1}{12}$ using only stability or linearity, not both, and without regularizing or specifying a summation method?”

Regarding the proof by Tony Padilla and Ed Copeland that $1+2+3+\dotsb=-\frac{1}{12}$ popularized by a recent Numberphile video [LINK] , most seem to agree that the proof is incorrect, or at least, is not sufficiently rigorous. Can the proof be repaired to become rigorously justifiable? If the proof is wrong, why does the result it computes agree with other more rigorous methods? Critiques of the proof seem to fall into two classes. One class of responses is to appeal to higher math for justification. For example, appeal to zeta function regularization [LINK] , as is done by Edward Frenkel in a followup video for Numberphile [this video] and a more recent video by Mathologer [LINK] , or to use an

Cites jimjim · 2017-12 · Mathematics Stack Exchange · on “Name of $\prod_{n = 1}^{\infty}n = 1 \times 2 \times 3 \times 4 \times 5 \times \cdots$”

You can call it Euler's nugget product , using the analogy from [Euler's Nugget]. 1 [this video] Dealing with divergent series or products is not as useless a concept as some might suggest, the divergent series and products do have uses e.g. see this [LINK]

Cites ntg · 2018-05 · Mathematics Stack Exchange · on “Why does an argument similiar to 0.999...=1 show 999...=-1?”

I agreed 100% with your answer(+1)... Then checked this video ( [this video] ) It seems there are mathematics where the sum might be -1, but not -1 as we know it, and not with the mathematics I know and understand... The argument is similar to " what is the root of -1"... If asked long time ago, one would explain it does not exist... at least for real numbers...

Cites Brian Faulkner · 2018-06 · Mathematics Stack Exchange · on “Can the hyperreals be used to describe the "gold nuggets" found with nonconverging series and Casimir forces?”

In one of Numberphile's videos [this video] they describe the $-\frac{1}{12}$ as a "gold nugget" inside the sequence $1+2+3+4+\dots$ surrounded by a bunch of "rock" that is infinity. Can these numbers that we get with regularization/analytic continuation of Reimann zeta/etc be described more simply using the hyperreals? Is there some particular non-arbitrary unlimited hyperreal $A$ you can associate with $1+2+3+4\dots = A-\frac{1}{12}$? Like in a Casimir force calculation problem, where the forces on both sides are infinite but differ by some finite value (related to the $-\frac{1}{12}$), is there some specific, non-arbitrary unlimited hyperreal value $A$ associated with the infinite part of that