Infinite Power Tower Equations Battle!
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- Опубліковано 20 чер 2024
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This is the part 2 of the Infinite Power Tower!
start: 0:00
Solve x^x^x^...=2 0:18
vs. Solve x^x^x^...=3 2:17
Domain & Range of the infinite power tower: 4:19
Brilliant: 7:38
Check out 3b1b’s video on the power tower puzzle • The power tower puzzle...
*The Satisfyingly Strange Journey to the Infinite Power Tower*
Part1: Solving x^x^3=2 vs. x^x^3=3 • they don’t teach these...
Part2: Solving x^x^...=2 vs. x^x^...=3 • Infinite Power Tower E...
Part3: Domain and Range of y=x^x^... • Domain and Range of th...
Part4: Why (cbrt(3))^(cbrt(3))^... converges to 2.4 and NOT 3? • Why it doesn't converg...
If you enjoy my videos, then you can click here to subscribe ua-cam.com/users/blackpenredpe...
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blackpenredpen,
math for fun
Hi all, thanks for the 500k subs! I will make a post of the winners of the secret giveaway tonight. After that I will be taking a break from YT.
Congratulations
I am the second because first are u so can u give me heart
How long?
How long break??
I hope you have a nice break!
Can't wait for part 3, I need that domain explanation!
It's in the description 😂
@@ganeshprasad9851 oh damn you right, thanks 👍
ua-cam.com/video/oy_kPyTstqk/v-deo.html
Nerd 😲
Will you explain why the domain is like that in another video?
It’s in part 3, see description for the unlisted videos
Interestingly, if you have any quantity of the form a^(1/a), where a is positive and real, the corresponding power tower will always converge, because if for all such a not equal to e, a^(1/a) < e^(1/e), since at x = e, x^(1/x) attains its global maximum. So, despite the fact z^^♾ = a only converges if a < e or a = e, the power tower with z = a^(1/a) does converge. It just does not converge to a.
wayt, how is this comment 5 days ago??
HOW 5 DAYS AGO????
It may not attain that maximum on an open domain, ie the extreme value theory might not hold. X=e is just a critical point
If the tower doesn't converge to a, it still has to converge to y such that y^(1/y) = x = a^(1/a), so for a>e, the tower converges to the unique number y (between 1 and e) such that y^(1/y) = a^(1/a). This is the same unique number y such that y^a = a^y. In bprp's example a=3 and y = e^(-productlog(-ln(3)/3)) ~ 2.48 where productlog() is the name used for the Lambert W function on WolframAlpha. Note that 2.48^3 ~ 3^2.48
thatnhoxiu The video was unlisted.
When are you going to post .. isn't your break over ? 😔
You're probably one of my fav youtubers of the last 2 years, bro! Your work is awesome.
I'm really passionate about maths and you just feed my hype! And in these days of quarantine and isolation, you and Dr. Peyam sometimes even feel like my "math friends"!
Hey listen; have you ever thought about a good video series about Stochastic Calculus? I'm quite into it right these days! (financial markets and stuff). Just learned how to derive Black-Scholes' PDE. But I get lost when certain subjects such as Ito's integral or similar come around. It would be excellent if you play some of that sort of things!
Long live BPRP!
Currently it's 2 am here, and I don't know why I'm watching this at this solemn night, but still I'm enjoying Idk why.
For anyone who is curious, if you want to know what x^^♾ is equal to for any x in the domain, then notice that y = x^^♾, and x^^♾ = x^(x^^♾) = x^y. Therefore, y = x^y.
To solve for y, take the natural logarithm here. We can do this because are assuming x is positive in this initial exercise. Hence ln(y) = y·ln(x), implying ln(y)/y = ln(y)·exp[-ln(y)] = ln(x), hence -ln(x) = -ln(y)·exp[-ln(y)]. If -ln(x) > -1/e or -ln(x) = -1/e, then the above equation implies W[-ln(x)] = -ln(y), where W is the Lambert W function, in this case, the principal branch of the W map. Therefore, y = exp(-W[-ln(x)]) = 1/exp(W[-ln(x)]) = 1/(-ln(x)/W[-ln(x)]) = -W[-ln(x)]/ln(x).
The condition that -ln(x) > -1/e or -ln(x) = -1/e implies that ln(x) < 1/e or ln(x) = 1/e, which implies that x = e^(1/e) or x < e^(1/e), which agrees with what was stated during the video.
Also, x < e^(1/e) implies ln(x) < 1/e, which implies -ln(x) > -1/e, which implies W[-ln(x)] > -1, which implies -W[-ln(x)] < 1. Therefore, -W[-ln(x)]/ln(x) = y < e, or equal to e if x = e^(1/e), which also agrees with what is stated in the video.
4 DAYS ago...
Субс Тим The video was unlisted.
Congratulations on 500k!
Here's to 500k more!
I really like your channel. It is way more entertaining than the other math channels.
Thanks !I wish you happy new year!
Managed to got half of bounds. Namely, x^x^x^...=y means x^y=y, this solves using Lambert W function: y=e^(-W(-ln x)) (I've seen your other videos where you explain what is W and how to solve such equations). But, W(x) domain is x≥-1/e (there are two real branches: W₀ domain is [-1/e; ∞), with values in the range [-1; ∞), W-₁ domain is [-1/e; 0), with values in the range (-∞; -1], that's a multivalued function). This means, there must be -ln(x)≥-1/e, solving that for x gives x≤e^(1/e). Then, if I put that value into equation, I've got y = e^(-W(-ln x)) = e^(-W(-1/e) = e¯¹ = 1/e.
Now I need to prove this is lowest possible value of y. Also still haven't figured out how to find out other bounds.
I feel, we need to go from the definition of x^x^x^..., consider series f₀(x)=1, f₁(x)=x^f₀(x)=x, f₂(x)=x^f₁(x)=x^x, and so on, f_n(x)=x^{f_{n-1}(x)}, then define "power tower function" f(x) = lim f_n(x) for n→∞. Now we have to ask ourselves if this ever converges, and if it does, for which values of x? And this shoud give us all bounds.
The solution x=f¯¹(y)=y^(1/y). Let's analyze this formula. It's derivative is df¯¹/dy = y^(1/y)(1-log(y))/y². For df¯¹/dy=0 we get y=e, x=e^(1/e). This is the maximum of the f¯¹(y), that's why when we put 3 in it we have some adequate value, which happened to converge our power series to something less that 3.
Awesome follow-up to your other video!
This is brilliant 😉
I can't believe you've never seen this problem? I first saw it in 1968 as an undergraduate!
Perfect followup.
Very good!
Great video!!
Thanks
Damn the video's real smooth ;D
I'm a math hobbyist, I don't have advanced education in math, but I love them and I'm able to understand most of your videos (they're awesome). I have a question due to my ignorance: When you have a tower exponent of real numbers, it has to be solved up-to-down? I mean, the solution you give to x^x^3 only works if we solve the exponents up-to-down, and then the same answer fits to any x^x^x^...^x^3 form. Is it correct? Thank you very much in advance!
Similar technique can be used for nested square roots/fractions equations.
Half a million subscribers !!! Nice job !
i forgot that this man was having a break
the maths is amazing . I love the infinity
hey i love your videos. Can you find the maximum and minimum values of f(x,y)= (x^y)/(y^x) by using partial derivatives?
500k hits.
Congrats
Hey, then what's the problem with tower(x) = 3 ? Is there any clear explaination without using the range of y?
Congratulations for got 500k subscribers
It's fun to see this video before it's published.
Now it has been published lol
How do you see these
ItIsLamin you can find them in playlists, he puts his unlisted videos in playlists before he publishes them.
@@integralboi2900 oh
adoro esses videos mesmo. matemática é a linguagem universal
Hey I want to ask u one question related to integration.While doing Integration of cos square x or sin square x why don't we use the simple linear Integration formula why we use formula of cos2x in doing their integration.please reply...
Gracias
Isn't the infinite power tower a kind of tetration where the "exponent" approaches infinity? And btw, could you please make a video on the inverses of this operation? Namely the super root and the super log.
And thanks!
Yes, an infinite power tower is "equivalent" to x^^♾.
Also, tetration to a fixed exponent only has one inverse map, in this case, the superlogarithm of infinite order, which is expressible in terms of the natural logarithm and the Lambert W function.
HOW 5 DAYS AGO
@@timka3244 they're probably members
Bprp do you think you can do a video about the Bessel's differential equation and it's series solution?
Suggestion: Instead of using a whiteboard or blackboard, use a transparent glass.
The setup would be you facing the camera and glassboard between you and camera. Obviously, camera will be recording all the writings backward. Then using software, convert/mirror the video.
This way, you dont have to keep turning your head to look at camera (to look at the audience); you will always be looking at the audience while writing the equations.
5:40
Engineering students don't know why, because π=e=3
Lmao
We are using e=2.7 actually.
The worker's of Babylon estimate π more accurately than engineering students now! LOL
We use e equals to 2.8 only
nice
so what is that number that the tower of 3^(1/3) converges to? How do you solve for the value of such an expression?
See part 4 in the description
Can you do a video on the Newton-Raphson method for solving something like 3x^4-7x-1=0 etc?
Hello, Mr.Cao, can you do a video on volume of revolution in polar coordinate without using double or triple integral?
So that's his name?
A monk carrying a shark🤙🏻🤙🏻
A question is this concept releted to the Mandelbrot set? If yes then how?
Also at 2:53 you forgot to add the doremon music
Really dissapointed!!
Great video tough.
Hey , if i have a polynomial,should its factors also must be a polynomial ? Please answer
In your comeback, can you make a video tutorial about other subjects like engineering courses dynamics, statics, strema etc? Im taking engineering courses and I enjoy your video about tips and tricks and you explain it very well than my prof.
showing the existence of x^x^...... is very important.
If x=a^(1/a) but the tower (x^x^x^x...) doesn't converge to a, it still has to converge to y such that y^(1/y) = x = a^(1/a), so for a>e, the tower converges to the unique number y (between 1 and e) such that y^(1/y) = a^(1/a). This is the same unique number y such that y^a = a^y. In bprp's example a=3 and y = e^(-productlog(-ln(3)/3)) ~ 2.48 where productlog() is the name used for the Lambert W function on WolframAlpha. Note that 2.48^3 ~ 3^2.48
I still dont understand it where it comes from,but the info is excellent
Hey Bprp, Can you please explain definite Integration of Greatest Integer function, it's very confusing
Respect
Hi I like your videos very much can u please explain through a video why derivative of lnx is 1/x
Hey, could you do the limit as n goes to infinity of (-2^n)sin(pi/2^n) this must converge to pi but i didn't find out how to find it yet
PS: I know the answer by using geometry with regular polygons of 2^n sides, and scalar product
hi, bprp! Wanna cool task? Just look at this: tg(sin(x)) or sin(tg(x)) which is bigger solve for x on interval (0; pi/100)? Have fun!)
Hi,bprp, can u explain why we can't integrate 1/dx
I got a idea. Since now you have 500k subscribes, do 500 integrals in one take to commemorate it. Who more agree with this? 😂
that's such a reckless idea. i have never heard it before.
but if you want, then split it into many short videos
Waooooo this is awesome
I like the chain chomp microphone
hey how are you? It’s been 2 months since you posted video last time. Are you OK?你还好吗？好久没看到你了！
Hey BPRP, I know you won't see this but I'm going to ask anyways...
How did you hold the camera directly above your paper in some of your older videos?
There's a stand for that if u want professionality in ur work
I used a huge pile of book for that LMAO😂😂😂😂😂
5:44 *technically speaking*
Ah yes, the engineering approximation.
I have a Q. What is infinite series of epsilon?
Hey bprp,
4:38 - 4:58 . . . How do you prove this?
It's in part 3, you can see the video in the description : )
Is the derivative of ln(x!) lnx?
How to find mirror image in co-ordinate system ,sir please
How to find rank of matrix and solution of system of linear equation,eighen value by using application of matrix,amd A inverse also
can you compute productlog(2)
The converging value is
x=W_0(a)/a
where W_0 is the principle branch of the Lambert W function and
a=-ln(cbrt(3)).
If I use W_(-1) instead, I get x=3 which does not make sense unless we somehow redefine convergence.
All numbers go past sqrt(2) is considered as infinite. Therefore two equations are equal.
Nice. But… why does this reasoning converges to sqrt 2 for tetration and not for normal power from bottom to top (in which 3^3^3 would be 27)?
How can 3^3^3 be 27? If you mean not as normal power tower (without parens it’s top to bottom), but left-to-right evaluation, it’s (3^3)^3 = 27^3 = 19683. The other way around it’s 7625597484987.
The sequence doesn’t converge to √2, but an infinite power tower (not tetration) of √2^√2^√2…. actually converges to 2.
Please do i^2!
Infinite Power Tower was not explicitly defined as limit of finite power towers, so you are free to define it another way, as lim of x from 2 (good value) to 3. This is called continuation
OCTAGRAM
Oh I mentioned about that in part 3.
Usual version I see don't say "you try (3^1/3)^... and get 2.4" but "you solve for 4 and get same result as 2"
This is an old problem I first saw as a teen in 1968!
Heyy when will you upload ur next video?
Sir, Rules of tetration say you must work "downward" from the highest exponent evaluating to the base. You are incorrectly working "upward", the wrong direction and will give a totally different value. How can the expression be evaluated if you can never start at the last exponent of an infinite tower power and work "downward"?
Hi Mark.
Whenever we are dealing with infinity, we should do it in terms of a limit. In this situation, we should define the inf power tower as the limit of a sequence of the functions x, x^x,x^x^x,...
For more details, please see part 3 in the description.
I am wondering: if you have an infinite power tower function f(x)=x^x^x^x..... you can write f(x)=x^f(x).
But can you also write f(x)=f(x)^f(x) ?
In the first case with x^x^x^x·.... you can never start evaluating the 'highest power' in the tower because the tower is infinite.
But the second case is even worse: how do you even start figuring out what (x^x^x^...)^(x^x^x^...) means? The first tower is already infinite, so how can I stack the second on top of it? Do I just get x^x^x^x^..... again?
If not, why not? I am confused.
Hey, make a video with the proof that the numbers with form abcd... = a! + b! + c! + d! + ... are finite. Ex: 145 = 1! + 4! + 5! It could be interesting
Thank..
Could you do the integral of sqrt(tan^-1(x))
At London's East End we call it an infinite paah taah.
Please!
Use your incredible knowledge to solve "root locus" problems!!!
Infinite power tower not to be confused with Tower of Power which is a Californian R&B band.
New video!
ua-cam.com/video/wIxDUPyUvk0/v-deo.html
Geometric integral!
Everything is going over my head🤐
Blackpenredpen should be change now into bluewhale
How to denote independent events on Venn diagram .
Please show the derivation of the range of y
Hi, can you explain again please why the infinite superpower of x can be written as x^2?
Yo
x^x^x^x... = 2
So because of x^(x^x^x^x...) = 2
We replace the never ending exponent with 2 as both are the same
So thats it, x²=2
@@nahrafe yh but because its infinite, doesn't that mean u could also write it as x^x^2 = 2, or x^x^x^2 = 2, and so on. Also when u tetrate sqrt2 to infinity, u don't get 2.
Look, It's steve, caveman style!
Is it possible to do the integration of tan(cosx)dx?
I mean from 0 to 2pi lol
2:03 ahh excuse me ......WHAAT
I’m having trouble solving the integral between 0 and inf of x^2 / (e^x - 1)
pls send help
Saludos :)
Hello teacher! Why don't you post about Integral more??
Please make a video on ith derivative of x^i
In the next video, can you solve the following integral?
Latex: \int _{-a}^a\sqrt{a^2-x^2}\;dx,\;a>0
What is your channel's profile photo about?
Plz since 4 years i wondered how integral 1/x=lnx can u make a video about it?
Not a proof but it’s kinda nice. twitter.com/blackpenredpen/status/1233863309562011648?s=21
@@blackpenredpen lol i see... u proved it with a method that only can be true when the whole thing is already proved...so there is no proof after all?
@blackpenredpen plz simplify:arcsin(x^2), x belongs [-1,1]
Hi bprp .. can u slove the integral for (x⁵+1)^-1? ..please i cant slove it ..
@blackpenredpen
What happens when you plug in complex values for x in an Infinite Power Tower?
Gets very complex, cannot be easily visualised
Who was the mathematician that proved those crazy thing?