LOG#178. Divergent sums: The Number awakens!

summary-divergencesAre you divergent?

Divergent1maxresdefaultDivergentdivergent-quoteDivergences are usually sums or results you would consider “infinite” or “ill-defined” (unexistent) in normal terms. But don’t be afraid. You can learn to “regularize” a divergent series or sum. Really? Oh, yes!divFactionsBeyond faction of a well-known book (and movie) series, we can do it. We can sum divergent sums!

Let me consider the sum S = \sum_ {k = 0} ^ \infty (-1) ^k = 1-1 + 1-1 + 1 -\cdots

You think perhaps in adds to zero like

(1)   \begin{equation*} S = (1-1) + (1-1) + (1-1) + \cdots= 0 \end{equation*}

Or, maybe, you even add it to obtain 1 in the following way:

(2)   \begin{equation*} S = 1 + (- 1 + 1) + (- 1 + 1) + \cdots = 1 \end{equation*}

However the mathematician (and some crazy theoretical physicts) have sinister ways of adding this sum you would qualify as “ divergent ”. `What create and do these crazy people of physics and mathematics? I will tell you. But I warn you. They are Dark Arts. Taking common factor:

(3)   \begin{equation*} S = 1- (1-1 + 1-1 + \cdots) = 1-S \end{equation*}

(4)   \begin{equation*} S = 1-S \rightarrow 2S = 1 \rightarrow S = \dfrac{1}{2} \end{equation*}

This seems (black) magic and delirious, because an infinite sum of alternating numbers providing  a fractional number seems to be really from another dimension or parallel universe. But under certain conditions it can be done … And worse, it serves to “ test” results yet more ​​disturbing. The following sum:

(5)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + \cdots = \sum_ {k = 1}^ \infty 1 = \sum_ {k = 1} ^\infty (-1) ^{2k} \end{equation*}

you can add using the above sum S = 1-1 + 1-1 + 1-1 + \cdots = 1/2, because if:

(6)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \cdots \end{equation*}

(7)   \begin{equation*} 2A =  2 + 2 + 2 +2+ \cdots \end{equation*}

(8)   \begin{equation*} A-2A = -A = 1-1 + 1-1 + 1-1 + \cdots = S \end{equation*}

and therefore, as we had ominously calculated that S = 1/2, we have the amazing result:

(9)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + \cdots = - \dfrac{1}{2} \end{equation*}

Awesome! But there is a “one more thing” … We will now calculate a sum to think it really gives infinite:

(10)   \begin{equation*} B = \sum_ {n = 1} ^\infty n = 1 + 2 + 3 + 4 + 5 + \cdots \end{equation*}

To do this, let’s do another trick of mathematical magician:

(11)   \begin{equation*} B = 1 + 2 + 3 + 4 + \cdots \end{equation*}

(12)   \begin{equation*} 4B = 4 + 8+ \cdots \end{equation*}

(13)   \begin{equation*} -3B=B-4B = 1-2 + 3-4 + \cdots = \sum_ {n = 1}^\infty (-1) ^{n + 1} n = C \end{equation*}

Now calculate, using past computations:

(14)   \begin{equation*} C = 1-2 + 3-4 + \cdots = \sum_ {n = 1} ^\infty (-1) ^{n + 1} n \end{equation*}

Our first concrete result, squared, can be rewritten as follows:

(15)   \begin{equation*} S ^2 = (- 1 + 1-1 + 1-1 + \cdots) ^2 \end{equation*}

Or well

(16)   \begin{equation*} S^2 = S \cdot S \end{equation*}

(17)   \begin{eqnarray*} (-1 + 1-1 + 1-1 + \cdots)\\ \underline{\times (-1 + 1-1 + 1-1 + \cdots)} \end{eqnarray*}

(18)   \begin{eqnarray*} =1-1 + 1-1 + 1-1 + 1-1 + \cdots \\ -1 + 1-1 + 1-1 + 1-1 + \cdots\\ + 1-1 + 1-1 + 1-1 + \cdots\\ \cdots \end{eqnarray*}

(19)   \begin{equation*} 1-2 + 3-4 + 5-6 + \cdots = C \end{equation*}

Therefore: S ^ 2 = C. As S = 1/2, then S ^2 = C = (1/2) ^ 2 = 1/4. Therefore, B = C/ 3 = -1/12. That is (to freak out a last time), we have shown that

(20)   \begin{equation*} \boxed{B = 1 + 2 + 3 + 4 + 5 + \cdots = - \dfrac{1}{12}} \end{equation*}

Astonishing! However, all this stuff is not new. It was known for many people before me, and it was Ramanujan and later Hardy in a book titled Divergent Series where you can find fine theorems about this crazy subject.

Other divergences:

  1. DC comics divergence series…


2. Divergence theorem!

div-div-thmA3. Futurama divergent series…

FuturamaDivergenceSee you in my next blog post!

P.S.: Moral…


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LOG#178. Divergent sums: The Number awakens! — 8 Comments

    • You’re welcome…You are the fourth person who passes my spam-bot test…Congratulations!!!!

    • #8 to pass my spam-bot test. On your question: in a formal DIVERGENT series that is completely right…You can not PLUG just A=1 because indeed is “a formally infinite divergent series”…As a formal series, equation (8) and others in this post are right.

      • the series are not the problem, but the terms:

        A= 1 +1 +1 +1 +1 +…..

        2A= 2+ 2+ 2+ 2+ 2+…..
        -A= 1-2+1-2+1-2+1-2+1-2+…..=1-1-1+1-1-1+1-1-1+1-1-1+1-1-1+1-1-1…

        Each term in the las series has the shape 1-2 and even grouping 1’s and -1’s I coudl not find a way to get 1-1+1-1+… without messing with the orther in the series (somethig dangerous in series)
        I will try to read the books you suggest but perhaps you should see


        I have to see re rest of the Carl Bender videsos on Mathematical Physics (I have only seen the 4 first) but at least this can help you in the next post

        • Dear reader: it is not ONLY the grouping, but also de PLACING or ARRANGEMENT of terms…Please, see this:

          -2A= -2-… -2 – -2-…

          You have to read Divergent Series by Hardy, also some classical books by Ramanujan (the chapter of Ramanujan’s notebook about sum of divergent series is a classic work, not the most rigurous, but the origin of this thread), and some cool Lectures by Chua where he also discusses this. There is no mystery in the divergent series theory. You must choose a procedure to sum partial sums and define some rules. Please, I am a theoretical physicist, mathematician at core, and soon I will be also astronomer and astrophysist… I do know what I am doing. Not a fair point to tell me what I should do…And by the way, series in Quantum Field Theory are also usually divergent, and only meaningful asymptotic series. I am well aware about the notion of convergence, regularization, renormalization and all that. Cheers.

          • you are getting a good formation but, Unless you were the master of the univer unable to fail, it is fair.I have read enough of your post to now it (most of them are good but some of them are a bit obscure ) Also it is fair form you noting that you know what are you doing, but pease, do not une authority as a reason in science.

            The problem in this post is not the result but the use of

            -2A= -2-… -2 – -2-…
            this is dangeous becaure you are not adding

            and the introduction of these zeros is not alwais JUstified (remember series 1 + 0 − 1 + 1 + 0 − 1 + …).
            I may be “no mystery in the divergent series theory” but they are quite tricky (EJ the same method you used with the 1-1+1-1… series may give 0=1 when directly applied to the unbounded series1+1+1+1…)

            You Are usiNg a method like the euristic method oF Ramanujan for the seRies 1+2+3+4…(http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm)

            that cAN be problematiC beacuse If the introductIon of infinite zeroS allon the series, as pointed out at

            https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF (euristiC sectiOn)

            Mean while I wilr read those clasic boocks and have a look to Ramanujan NoteBooks and the article
            ad soon as I can

          • Dear reader.
            The manipulation of divergent series is tricky, no doubt about that. But you can arrange and place terms to get any result you want, essentially. For instance, I suppose you know that there are several classical series and summation methods, so there is no mystery about why we can get different values (e.g. Cesàro summation). However, some methods of summation, like that using the Riemann zeta function, show that some “regularizations” make more sense than others…That the heuristic method of Ramanujan equals to the Riemann zeta function is a deep result. Ramanujan himself had more powerful summation methods using theta functions I introduced in some posts. Today, that method has been generalized to very sophisticated methods called 1-psi-1 summation and higher dimensional generalizations based on hypergeometric functions. I have a post about it for a future entry…If you don’t disturb with such a garbage…

            About my formation and “authority”. So you don’t value authority…OK, noted… I am not going to discuss this point too much but…since… After all, I am a scientist and teacher…Well,…To be more precise… I am not claiming my formation implies what I say is true (note the point and I did NOT say that in any moment, did I?), Science does not work that way of course. I only point out that my formation (even if I am or I am not a Ph.D) gives me the right to say that I know what I am doing (or should you banish the opinion of experts?). I think it is fair to say that…I am not being selfish but only balancing my knowledge with the RIGHT to give an “expert opinion”. I presume…Or should you value more the opinion of someone who do physics, mathematics, chemistry with other exotica (some people exist out there unfortunately) but has no formation in Science but it is just crackpottery stuff? If you prefer Deepak Chopra or any other pseudoscientist, go away from my blog… And before doing any further comment I do know about divergent series and their axioms or features, read a bit more…I am well aware of everything you have said. Cheers. PS: If you continue with this offensive attitude I will block you, delete all your comments and report you to WP (and I know who you are already). I write this blog with passion for me and my readers, not for people giving poor arguments or stuff they can find in the literature.

            PS(II): Also, read this, https://en.wikipedia.org/wiki/Ramanujan_summation

            PS(III): Albert Einstein — ‘To punish me for my contempt for authority, fate made me an authority myself.
            Albert Einstein — ‘Blind belief in authority is the greatest enemy of truth.’ I suppose you will not agree on authority, it seems, but I am not asking BLIND belief in my authority (indeed, I have no one but the logic of mathematics and the principles of the scientific method). I agree authority IS NOT a proof of truth, but authority and ability allow us to make remarkable arguments.

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