Ramanujan: Making sense of 1+2+3+… = -1/12 and Co.

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The Mathologer sets out to make sense of 1+2+3+ … = -1/12 and some of those other notorious, crazy-looking infinite sum identities. The starting point for this video is the famous letter that led to the discovery of self-taught mathematical genius Srinivasa Ramanujan in 1913 (Ramanujan is the subject of the movie “The man who knew infinity” that just started showing in cinemas.) Find out about how these identities come up in Ramanujan’s work, the role of “just do it” in math, the rules for adding infinite sums on Earth and other worlds, and what all this has to do with the mathematical super star the Riemann Zeta function.

You can download the jpeg of Ramanujan’s letter to Hardy that I put together for this video here:
(quite large)
You can access a scanned copy of Ramanujan’s notebook here:
or check this out

If you want to watch some other videos that deal with these strange identities I recommend the following:
(a Numberphile video featuring the mathematician Edward Frenkel who is also talking about the connection between the Riemann Zeta function and Ramanujan’s crazy identity.)
(James Grime’s video (singingbanana & Numberphile) which aims at making sense of the formula at the heart of Ramanujan summation which I only flash briefly in the last part of the video.)
(a very nice 3Blue1Brown video that makes sense of the identity 1+2+4+8+ … = –1 in a way that is totally different from the one I am talking about in this video.)

This week’s video is in response to a large number of you and my students at university asking me for my take on the whole 1+2+3+… =-1/12 business. Initially, the plan was to come up with one of my usual 15 minute long videos. However, after several unsuccessful attempts at not exceeding the magic Mathologer time limit, I realised that any “short” video like this would just be a clone of one of those wrong/misleading accounts of this topic that YouTube is full of. So, rather than just give up completely on this project I decided to do what you should never do if you actually want people to watch your videos, namely simply go for it and not look at the clock. The result – a video that is an insane 35 minutes long in which I say all the things that I think need to be said and can be said only using elementary math to (just barely) do this amazing topic justice.

Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy 🙂

Burkard

P.S.: If you know calculus and want to read up on all this some more, beyond what is readily available via the relevant Wiki pages and other internet resources, I recommend you read the last chapter of the book by Konrad Knopp, Theory and applications of infinite series, Dover books, 1990 (actually if you know German, read the extended version of this chapter in the original 1924 edition of the book). People usually recommend Hardy’s book, Divergent series, but I’d say only look at this after you’ve looked at Knopp’s book which I find a lot more accessible. The Hardy who wrote this book is the Hardy who I mention at the beginning of the video! Having said that, Hardy’s book does have quite a bit of detail on how Ramanujan summation applies to the Zeta function; see chapters 13.10. and 13.17.

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21 COMMENTS

  1. Недавно читал, что его тетради «пропали». Я читал книгу «Гиперпространство».
    Этого индийца многие упоминают, когда рассказывают про другие многомерные пространства… Мне кажется, что его источники где-то в старинных каких-то манускриптах были и простым людям не доступны.Потому что это довольно сложно всё самому понять.
    Спасибо за видео.

  2. *The man who knows Infinity – Ramanujan
    The man who brought the concept of Zero – Aryabhat

    Both are from India.
    Proud to be an Indian. Jai Hind, jai Bharat…..*

  3. whenever such a sum appears anywhere in physics – I explicitly emphasize that not just in string theory, also in experimentally doable measurements of the [Casimir force] (between parallel metals resulting from quantized standing electromagnetic waves in between) – and one knows that the result is finite, the only possible finite part of the result that may be consistent with other symmetries of the problem (and that is actually confirmed experimentally whenever it is possible) is equal to −1/12.

    It's another widespread misconception (see all the incorrect comments right below your question) that the zeta-function regularization is the only way how to calculate the proper value.

  4. Сумма целых чисел или их целых степеней никогда не даёт дробное число.

  5. commented yewars ago but rewatching it how comes I did not post the old infinite Sum joke? "an infinite number of mathematicians enter a bar: the first orders one pint, the next half a pint, the next a quarter pint the one after an eighth pint the one after that orders one sixteenth of a pint and the barman loses patience. He shouts "i´ve got enough of this" and slams two pints on the table"

  6. Just you would think that if I get my test what does the sum of all positive integers I would write divergence as answer. No sir, you go there and you write -1/12 and you take your 0 points as the math enthusiast you are!

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