Can you show some finite and clear exact representations of π?

Sure! In 1841 Karl Weierstrass [ https://en.wikipedia.org/wiki/Karl_Weierstrass ] defined [math]\pi[/math] directly as an integral: [math]\quad\displaystyle\pi=\int_{-\infty}^{+\infty}\frac{\te…

Top stories for Long   Can you show some finite and clear exact representations of π? Alan Bustany, Trinity Wrangler, 1977 IMO • Answered March 20, 2018 Sure! In 1841 Karl Weierstrass defined [math]\pi[/math] directly as an integral: [math]\quad\displaystyle\pi=\int_{-\infty}^{+\infty}\frac{\text dx}{1+x^2}[/math] If you want finite bounds on the integral you can use: [math]\quad\displaystyle\pi=\int_{-1}^{+1}\frac{\text dx}{\sqrt{1-x^2}}=2\int_{-1}^{+1}\sqrt{1-x^2}\text dx[/math] Or you can use trigonometric functions: [math]\quad\displaystyle\pi=4\arctan1[/math] Or other functi... Read more » 213 16   Have you ever seen an employer fire someone without realizing what a crucial role the employee played? Bug-Z • Answered March 2, 2025 I once saw an employer play "lay off a whole department and replace with cheaper employees" with every department. They kept the occasional employee who was willing to deba... Read more » 576 21   Is a creatinine level of 1.21 dangerous? Theodore Kirsten, Mastermind (2009-present) • Answered January 20 I am a 54-year-old male. My creatinine level was 1.4, and I managed to decrease it to 1.0 with lifestyle changes within one month. Alcohol intake reduced from the baseline o... Read more » 221 12   Why did Perl lose to Python? Dale Gulledge, Real time software developer for mumble years • Answered November 3 As someone who used to user Perl a lot and now uses Python, I can tell why I switched. Perl 6 took an incredibly long time to arrive. It was almost the mirror image of what ... Read more » 95 17   Why should CPU and RAM be separated? Why can't we combine them into one "entity" knowing that the CPU is much faster? Brett Bergan, Building PC's for 30 years • Updated February 22, 2021 I finished building my first PC in December of 1996. It had 16MB of RAM. I remember one Saturday morning staring at my plywood bookshelf thinking how awesome it would be wh... Read more » 1.2K 79   Have you ever confronted a seat kicker on a flight, and what ended up happening when you did? Bob Wilson, Flew 20 times a year for 20 years. • Answered January 18 Yes I did, it was a boy about 4. I leaned over the seat and very calmly told him that it bothered me when he did that. He completely ignored me, and kept kicking the seat w... Read more » 4.8K 203   My landlord hasn't picked up rent for almost 1 year. He doesn't answer the phone or text. What could happen? Shane Conneely • Answered June 24, 2025 Something similar had happened some college friends of mine. Their landlord had gotten divorced and had moved to Saudi and then he just vanished. The bank account that they... Read more » 593 71   Does marijuana lower or increase blood pressure? Brian Butler, former Combat Medic at Retired US Army Combat Vet • Answered March 11, 2024 I am a retired medic. I have high blood pressure and I have been doing my own little research and study on cannabis and blood pressure. So far I have not taken any blood pr... Read more » 1.6K 220   Would a 20mm machine gun be effective against infantry? Roland Bartetzko, Logistics in Ukraine (2022-present) • Answered February 7 I was a 20mm gunner in the German Army on three different platforms: the "Luchs" armored reconnaissance vehicle, as well as the "Kraka" and the "Wiesel" airborne weapons ca... Read more » 1.2K 52   How true is it that saying "arigato" at a store would be acceptable for a foreigner but rude for a native Japanese? Paul Richter, lives in Tokyo • Answered February 4 I assume this about the customer saying "arigato", after the purchase has been completed and the salesperson has thanked the customer. It wouldn't be rude if said by anybody... Read more » 296 14   Why is factoring numbers into primes a difficult problem? Alon Amit, CS degree and many years of coding. • Updated July 28, 2015 I must, sadly, disagree with the posted answers that point to the chaotic distribution of primes as the reason. This has nothing to do with the problem of factorization. Fi... Read more » 743 35   Which actresses aged like a fine wine? Peeyush Bhatnagar, I have read about the life of many celebrities. • Answered January 21 I personally feel Salma Hayek has cracked the code to aging gracefully. Over the years, it feels like she is somehow aging in reverse. She credits her youthful appearance, e... Read more » 2.2K 132 Read more in your feed

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