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

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.

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 instead of 'the blind leading the blind' (you as a 'blind' teacher, leading you as a 'blind' learner). You have the efficiency of being led by a mentor of some kind that can steer you away from faulty concepts that may come in. It's great that we now have more free/cheap materials than ever before at our disposal but without a mentor or some kind of peer-review, we could be misapplying concepts. Also, to comment on something you specifically said: > Because in my companies view, being able to teach yourself all that math is much more impressive than being taught from a University. Yes, it's 'impressive' but...most don't learn this way. Which is way it's 'impressive'. Also, being self-taught, how do you truly verify what you understand mathematically is accurate and solid? [2] You might be an