1 2
133 134
Posted: 6/22/2026 1:38 PM
Anonymous3892
Player (43)
Joined: 1/1/1970
Posts: 880
Lucky TAS of 2011 Gameboy TAS of 2011
Let me give a more detailed discussion about my question:
p4wn3r wrote:
Find the error in my proof that 0 = 1. Let Use integration by parts to find Now, with the change of variable x=uL, evaluate it like: Therefore, 0 = 1.
The error is in the next-to-last equals sign. As Denial140 pointed out, it's an exchange of limits that was not justified. Formally, we have a family of functions f_L(x) on [0,1]. We know that lim f_L(x) = 0 on [0,1] as L goes to infinity. However, the left hand side of the equals sign is lim_{L to inf} integral of f_L(x), and the right hand side is the integral of lim_{L to inf} f_L(x). Is the limit of the integral the same as the integral of the limit? In most cases, yes, but we'd better check the conditions here. The result that's used a lot for these checks is the dominated convergence theorem. Essentially, what the theorem says is: if all the functions f_L(x) on [0,1] are bounded above by an integrable function g(x) on [0,1], which integrates to a finite number, then yes, the two limits are the same. That's exactly where the problem is, here. It turns out you can calculate the envelope of this family of functions. I won't show the derivation here (it could be another question :D), but it's y = 4/(x*e²). Let's see it on desmos: It should be clear why the curve converges to zero at every point, but the limit of the integral does not go to zero. As L grows, the function dies off at most of the interval, but becomes concentrated at a peak that gets closer and closer to zero, and contains the entire integral. Also, since the integral of the envelope function y = 4/(x*e²) is not finite on [0,1], we cannot apply the dominated convergence theorem. Therefore, the exchange of limits is not valid, and it won't surprise many people that 0 is different than 1 :)
Posted: 14 days ago
Dacicus
Editor, Player (86)
Joined: 6/22/2005
Posts: 1065
Here's an algebra one: Let ϕ be the Golden Ratio (1 + √5)/2. Let F(n) be the n-th Fibonacci number: F(1) = F(2) = 1 and F(n) = F(n - 1) + F(n - 2) for integers n >= 3. Prove that F(n) = (1/ϕn - 1) Σnk = 1 (-1)n - k ϕ2(k - 1) for all integers n >= 1.
Current Projects: TAS: Wizards & Warriors III.
Posted: 2 days ago
Noxxa
Any
Moderator, Expert player (4308)
🇳🇱 Netherlands
Joined: 8/14/2009
Posts: 4155
Location: 🇳🇱 Netherlands
Exotic platforms TASer of 2018 Exotic platforms TAS of 2018 Exotic platforms TASer of 2017 Exotic platforms TAS of 2017 Exotic platforms TAS of 2017 Exotic platforms TASer of 2016 Exotic platforms TAS of 2016 Gameboy TASer of 2013
A few weeks ago, a couple irl friends introduced me to a weekday date guessing game that one of them developed: https://gilliver.co.uk/weekday-trainer It's a very simple game: you get given a random date of current year, and have to select what day of the week it is. Get 10 correct in a row, and your average time is recorded. There are some interesting methods to determine what weekday a given day falls on, called the Doomsday algorithm. After playing it on and off for a week or so on my phone while on the train or on lunch break (would definitely recommend over doomscrolling), I got a PB of 1.84 s average guess time. Curious to see if anyone else can get a good time on this? :o
http://www.youtube.com/Noxxa <dwangoAC> This is a TAS (...). Not suitable for all audiences. May cause undesirable side-effects. May contain emulator abuse. Emulator may be abusive. This product contains glitches known to the state of California to cause egg defects. <Masterjun> I'm just a guy arranging bits in a sequence which could potentially amuse other people looking at these bits <adelikat> In Oregon Trail, I sacrificed my own family to save time. In Star trek, I killed helpless comrades in escape pods to save time. Here, I kill my allies to save time. I think I need help.
Posted: 2 days ago
Randomno
Editor, Experienced player (598)
Joined: 2/11/2018
Posts: 284
Flash TASer of 2023 N64/DS TASer of 2021
Used to know a guy who did this. Never learned it myself though. Link to video
Posted: 2 days ago
Masterjun
He/Him
Site Developer, Skilled player (1913)
🇩🇪 Germany
Joined: 10/12/2010
Posts: 1192
Location: 🇩🇪 Germany
Exotic platforms TAS of 2020 Speedy TAS of 2018 PC TAS of 2018 Glitchy TAS of 2015 TASer of 2014 SNES TASer of 2014 TAS of 2014 SNES TAS of 2014 Funny TAS of 2014 Glitchy TAS of 2014 TASer of 2013 SNES TASer of 2013 NES TAS of 2013 SNES TAS of 2012 Funny TAS of 2012
Nice, incidentally I learned about the doomsday algorithm thing a few days ago. So I refreshed myself on it, and got an average time of 12s. But I'm more happy that I got them all right! :D I probably wouldn't have tried if one had to guess specific years, too. That seems more difficult heh.
Warning: Might glitch to credits I will finish this ACE soon as possible (or will I?)
1 2
133 134
1787809913