Ramanujan: Making sense of 1+2+3+... = -1/12 and Co.
Why Hacker News recommends it
Recommended as a careful, honest treatment of the -1/12 claim, often preferred over flashier videos.
What it is
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 by 3 HN commenters across 6 different months, 2016 to 2024.
What Hacker News said
Strong rec gus_massa · 2019-08 · on “The Ramanujan Summation: 1 and 2 and 3 and ⋯ + ∞ = -1/12”
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 l
Recommends stan_rogers · 2020-02 · on “Ramanujan Surprises Again (2015)”
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]
Recommends gus_massa · 2020-11 · on “Ramanujan's Easiest Formula”
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.
Recommends gus_massa · 2022-04 · on “I stopped working on black hole information loss”
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.
Recommends thaumasiotes · 2023-12 · on “Code is run more than read”
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
Recommends gus_massa · 2024-09 · on “Ask HN: What are some interesting math and physics blogs you follow?”
Not sure if this is what you want, but I like: The post of Mathologer about 1+2+3+...=-1/12 , probably [this video] The post of Minute Physics about the Higgs Boson , probably [LINK]
Cites CarolineW · 2016-07 · on “Does 1+2+3+... Really Equal -1/12?”
I think there's something wrong with the link you've provided: > Our systems have detected unusual > traffic from your computer network. > Please try your request again later. Perhaps this is the video you intended: [this video]
Critical heimatau · 2016-09 · on “How to Learn Advanced Mathematics Without Heading to University – Part 3”
I'm glad you had success but...let's not measure your outlier experience with the rest of the world. Especially in Adv Mathematics. I'd point you and other HNs to Srinivasa Ramanujan. He is self taught but...he was wrong [1]. He had a brilliant mind but...due to being self taught, he made some critical mistakes. Being self taught can easily lead the learner to some critical mistakes. Eventually, they may be corrected (and at what 'cost' does this mistake cause an organization or business or those involved) but it's more efficient of someone's time to just learn from another. I'm not saying everyone needs a University Degree. I'm saying that everyone needs a teacher. Everyone. Why? Because in