Mathologer
3 HN recommenders · recommended in 8 months, 2016–2024
34:25 · 3.6M views · Apr 22, 2016
Mathologer examines how identities like 1+2+3+... = -1/12 arise in Ramanujan's work, starting from his 1913 letter to Hardy. The video explains the rules for summing infinite series and why such results are meaningful but not naive sums.
Recommended as a careful, honest treatment of the -1/12 claim, often preferred over flashier videos.
8 comments from Hacker News
It's more complicated. The lie is using the same symbol for the usual summation and the extended version of the summations. We all agree about the value of a finite summation. We all agree about the value of a sum when the summation is absolutely convergent. We all (almost) agree when the summation converges, but the convergence is not absolute. This is the standard definition that is used in a calculus course, but be aware that when the convergence is not absolute there are some minor problems here and there. It's much better if you have absolute convergence. After that, the idea is that you can extend the definition of summation. The more easy way is [LINK] but the other comment posted a link to a more generic article [LINK] Cesàro summation is fine, you don't get too many surprises using it. But when you extend too much the definition of summation you get more weird results, and you loose some properties of the standards summation, for example a+b+c+d+e+... = a + (b+c+d+e+...). In particular, to get a finite result for 1+2+3+4+... you have to use extend the definition of summation a lot, perhaps too much, and there are many ways to do the extension, and not all of them give the same result. If you have half an hour to spare, I highly recommend to watch " Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. " by Mathologer this video
The result isn't owed to the zeta function. For example, Ramanujan derived it by relating the series to the product of two infinite polynomials, (1 - x + x² - x³ + ...) × (1 - x + x² - x³ + ...). (Ok, it's the square of one infinite polynomial.) Do that multiplication and you'll find the result is (1 - 2x + 3x² - 4x⁴ + ...). So the sum of the sequence of coefficients {1, -2, 3, -4, ...} is taken to be the square of the sum of the sequence {1, -1, 1, -1, ...} (because the polynomial associated with the first sequence is the square of the polynomial associated with the second sequence), and the sum of the all-positive sequence {1, 2, 3, 4, ...} is calculated by a simpler algebraic relationship to the half-negative sequence {1, -2, 3, -4, ...}. The zeta function is just a piece of evidence that the derivation of the value is correct in a sense - at the point where the zeta function would be defined by the infinite sum 1 + 2 + 3 + ..., to the extent that it is possible to assign a value to the zeta function at that point, the value must be -1/12. this video is a youtube video (Mathologer) which goes over this material fairly carefully.
It is weird, very weird. A very nice explanation of the proof is in a video by Mathologer " Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. " this video [spoiler alert] You can extend finite summations to infinite summations. The first extensions are nice and you get intuitive results, and are the extensions studied in Calculus in the university. But these extensions are not enough for this sum. You must make more bold extensions, and the results are not as intuitive, and not useful outside some special applications.
I agree that most videos in YouTube simplify the problem too much or are just wrong. My recommendations anyway: * Mathologer: " Ramanujan: Making sense of 1+2+3+... = -1/12 and Co. " this video * Vsauce: " The Banach–Tarski Paradox " [LINK] Each one is like half an hour long, but you may need to pause them and rewatch them a few times to understand the details.
Burkard Polster's (Mathologer) videos on the subject are probably going to be more useful than Numberphile. Numberphile merely presents a trick; Mathologer points out both that it's patent nonsense [0] and that it's useful patent nonsense [1]. [0] [LINK] [1] this video