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Curated by: Mathemaniac (96 videos)
Without differentiation or integration, or number theory, could we still prove this infinite sum? | Visit http://brilliant.org/Mathemaniac/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription. π/4 = 1 - 1/3 + 1/5 -..., also called the Leibniz series, is quite famous, but the usual proof involves differentiation or integration. The more visual geometric proof still relies a lot on some advanced theorems from number theory, but given how simple the series is, is it possible to have an even simpler proof? Yes! And this video tries to explain this. Actually, similar to the previous proof, this proof has been at the video ideas list for quite a long time - so it's good to finally see this out! By the way, on second thought, this looks similar to Fourier coefficients, at least the time average bit - though I can't see whether this is the same proof as the one using Fourier series of sgn(x). It feels very connected, but also very different in the sense that we don't need to use the Cesàro sums of the Fourier series. Please let me know if you have any ideas regarding this. Video chapters: 00:00 Intro 00:45 Chapter 1: The setup 03:16 Chapter 2: The main argument 14:55 Chapter 3: Making this rigorous 17:33 Sponsored segment This video was sponsored by Brilliant. Sources (in order of appearance in the video): (1) Pi hiding in prime regularities (3Blue1Brown): https://www.youtube.com/watch?v=NaL_Cb42WyY (2) Fermat’s Christmas Theorem (Mathologer): https://www.youtube.com/watch?v=00w8gu2aL-w (3) The proof itself: https://www.quora.com/How-can-I-show-that-frac-pi-4-1-frac-1-3-frac-1-5-frac-1-7-cdots-perhaps-geometrically/answer/Sridhar-Ramesh (4) Cesàro sums: https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation A little more reading: (1) Calculating pi by hand (Matt Parker): https://www.youtube.com/watch?v=HrRMnzANHHs (2) Regular sum, if defined, equals Cesàro sum: https://math.stackexchange.com/questions/1134265/cesaro-summable-series https://math.stackexchange.com/questions/514802/convergence-of-series-implies-convergence-of-cesaro-mean (3) Wikipedia proof (involving integration): https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 (Worth noting that this requires a bit more than just plugging in x = 1 in arctangent series - this requires the justification from Abel’s theorem) (4) If you want to prove the Cesàro sum of the positions to be 0 more rigorously, here is the justification: https://www.wolframalpha.com/input?i=sum+from+k%3D1+to+n+%281-k%2F%28n%29%29*cos%28kx%29 and https://www.wolframalpha.com/input?i=lim+n+to+infty+sum+from+k%3D1+to+n+%281-k%2F%28n%29%29*cos%28kx%29&assumption=%22LimitHead%22+-%3E+%7B%22Discrete%22%7D (and the formula in the first link would only require repeatedly applying product to sum formulas and telescoping series - still no differentiation / integration involved. If you want even further assistance, I made a video on this: https://www.youtube.com/watch?v=DzhK3UJj5IY ) Music: Asher Fulero - Beseeched Aakash Gandhi - Heavenly, Kiss the Sky, Lifting Dreams (From YouTube audio library) Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk For my contact email, check my About page on a PC. See you next time!