The Hardest Integral From The Hardest Test (Putnam Exam)
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- เผยแพร่เมื่อ 18 ต.ค. 2024
- 🎥 The Putnam's Toughest Integral Solved! 📚🔥
Join me as I tackle one of the most challenging integrals from the Putnam Exam: the infamous integral x^-x ! In this video, we'll break down the problem step-by-step. Whether you're a math enthusiast, a student preparing for the Putnam Exam, or just curious about advanced calculus, this video is for you!
🔍 What You'll Learn:
The Putnam Exam and its significance
Different approaches to solving integrals
Detailed solution to the integral x^-x
Tips and tricks for solving complex math problems.
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Thank you for watching, and happy solving!
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4:10 How do you know the argument to ln is larger than 0?
i think thats the best video for an introduction to your channel i could get
I'm glad you like it :)
I want to say the sammmee❤
@@improbabilty
You are a gentleman and a scholar, sir. Apology accepted. I personally don't want music to distract my thought process when doing math.
Absolutely amazing solution.
I appreciate that
that guy definitely had anger issues haha
People don't have ability to make videos like this TH-camr so they think that using such great words makes them powerfull.😂 Actually people who lack vocabulary start using such words.
Whatever you do people have some reason to scold you
Yes you better not to make him angry
The final value is roughly 1.3
Should be transcendental due to Liouville's theorem
1.29129 actually which is also pretty elegant
It's 1.29128599706....approx I graph it on desmos. Lol
@@zachariastsampasidis8880 common sense bro 😗
NIntegrate[x^-x, {x,0,1}] = 1.29129
that comment part felt more like a chess review
There is exactly one "k" in the video- at 12:58. I was able to follow everything easily because you said everything as you were writing. I just point it out because I thought it was funny that one time you realized the "k" looks like a "u" and made a special effort to make it look good, but immediately reverted back after that. Everything is good.
On a separate issue, I don't doubt that this is the solution that would get 10/10 points on the exam, but to me it doesn't look any simpler than the original integral. They are both infinite sums that have to be evaluated numerically. The solution converges quickly and even the first handful of terms are good enough to approximate the entire sum, so that's good if you were trying to get a numerical approximation. My problem is with the statement of the problem- it could have said show that the integral equals the sum.
There is one at 18:36 too
😭😭😭😭 I think this is because it's just been exam season so I got used to just writing really fast and not caring about handwriting
Sophomore's dream.
And true to life, I proved it (and the corresponding integral with x^x) in my sophomore year.
whys ur comment section filled with weird ppl ? ur videos r very well explained and ur voice is very calming :)
@@awkwardhamster8541 dats y we here
When I first saw the problem, I was tempted to use the definition of the integral as the limit of the sum as the index goes to infinity. It’s nice to see at least that the concept of integration is the limit of an infinite sum of a discrete sum.
I love the "complaining and explanation" that people seem to complain about. Just keep doing your thing lol.
Very nice solution btw.
Thanks mate !
I wish his penmanship were a little better.
Nice video but i thought it could use more music to jazz it up
Maybe a lot of people would have downvoted if the music was annoying
Yea we need to jazz this vid up!!!!!!
Would really help
Yes, thank you, a most surprising result.
Nice. I think Michael Penn did one on this integral too.
It looks like it would converge extremely quickly too, which would make it easy to approximate to a chosen level of precision.
I came for the math, but I enjoyed the analysis of dashxdr’s comment the most
Seeing limited integrals as the infinite sum of a converging discrete sequence helps conceptualize the process incorporated in this solution. After all, the only difference between the final left and right hand in the answer is the shift of the integration symbol to the summation symbol and the change of the limits of integration and summation (from zero to infinity) to (one to infinity) respectively. That makes perfect sense because X to the power of negative X has undetermined value when X = 0 or = 1 by definition. Of course, the original problem was written in terms of X not alpha, so the back substitution needs to be made.
Exactly what I'm thinking; back substitution to return the result in terms of the original variable x
how would a student know when to stop? The final result looks like an intermediate result as well.
Intuition and experience. This isn't an entrance exam or something where enough study is supposed to allow anyone with sufficient time to ace it. It's the Putnam exam, an extremely prestigious and entirely optional endeavor.
It’s relatively evident that you can’t simplify the sum. If not you probably don’t have the level to pass this exam
@@LouisLeCrack are you sure a more elegant expression doesnt exist "at your level"
The series converges very fast, therefore numerically useful.
awesome vid! now i just have to go learn about the gamma function lol
im new but i think your voice is really great!!!!
no music is no problem!!
Thank you, how nice :)
@@Jagoalexander you're welcome 🤗
you got a new subscriber and few views
You're the perfection of passive agressive lmao
I was thinking how the contrast between the thanks for all the constructive comments and the selected comment cussing him out was a textbook example of British dry humo(u)r. Chef's kiss.
Absolutely brillant ! Thank you a lot
Glad you liked it!
With luck and more power to you.
hoping for more videos.
The final result is mind blowing great work.😊
Thanks mate
Woa very cool result. Awesome vid
Given the behavior at x = 0, we can't integrate from exactly 0. We need to use a limit:
lim(ε→0+) ∫(ε to ∞) x^(-x) dx
I liked the music in your last video. I didn't think it was too loud. I thought it was quite calming.
I, however, did not like your choice of indexing variable in your final answer in this video. Why not just pick n?
Great video - very clearly explained. Keep up the good work.
I thought the music was incredibly beautiful
I COULD BARELY HEAR THE MUSIC OVER YOU SOLVING THAT INTEGRAL. SO OBNOXIOUS!!!
I know I'll be quieter next time
At x = 0, we have 0^0, which is undefined.
An awsome video it is! I wish your k and u (and possibly n) would be more distinctive.
You need a justification for interchanging the infinite sum and the integral at 6:50.
The series is uniformly convergent on [0,1], so the interchange of the sum and integral is OK.
@@attica7980 gotta prove that as well 😂
@@bantix9902 x ln x has absolute value less than 1 in [0,1], and the sum of 1/n! is convergent; hence the function series is dominated in absolute value by a convergent numerical series, so it is uniformly convergent.
There is a shorter path between these forms because both are equal to the limit as m goes to infinity of (sum k=1 to m ( ((k/m)^(-k/m))/m ). The 1/m represents the width of each segment of a partial sum in the integral.
Absolutely incredible. I love those Integrals. Thanks! I‘m surprised that I could solve this with my High school knowledge. :D
Great job!
Amazing video thanks a lot. I don't know if you know it but there is another integral that's look basically the same called "bernoulli integral", with the only deffencies being the negativ sign in the power. (No worries about the music we all make mistake 🫡🎉)
That's funny. I solved this problem while in my second year of undergrad. I was at my friend's place and we had ordered pizza. He had just finished taking his first course on Calculus and I joked that he couldn't find the derivative of x^x, which he did. Then he challenged me to find the integral, which I did too. It was fun!
What does the final summation converge to, e.g. according to Wolfram? Very nice video.
Ingenious! I wonder if the serial is convergent and if yes, what's the final result?
I knew the final equation in the video about the integral being equal to the sum. But how does that answer the question as to what the integral is. It would be good to know how the question was asked, since as stated (find the integral), one cannot know whether the final equation indeed solves the problem.
I have encountered this question before in a test in India. In that question, they asked the integer value less than or equal to the integral of x^x. So, you had to derive this result (we used reduction because gamma function isn't taught at high school level) and then calculate the first 4 terms to get the approximate value as 129 or something like that
Easy, full series and substitution
great vid!!!
Thanks!
Where’s the Family Guy and the Soap Cutting videos on the sides?
Please consider adding them next video!
I hope this is a joke 😂
Nice! What mic and software do you use btw?
Goodnotes on iPad and my airpods
Is this actually a Putnam question? What's the problem statement? It's a very cool result but how would a Putnam test taker know that the answer is simplified enough at the end, since it can't be evaluated exactly?
The question was show the integral can be written as the sum
Integral is a continuous sum , we solve it by finding the value of that sum but you transfer it to a discreet sum
amazing!!
Thank you for the excellent demonstration and...NO MUSIC is perfect so one can really concentrate🙂😂
Nice final formula
I loved the video and everything, but there's one thing that I would like clarification about. Isn't there any condition on the function that needs to be met in order to interchange the integral and the summation? And if so, was it satisfied here?
Thank yih
Check my latest video, where I go into detail about this
I Solved the Impossible Bernoulli Integral!
th-cam.com/video/T-1fFR8Nk6A/w-d-xo.html
Bro had me so confused with the u’s and k’s looking the exact same
Thank you for the video. Question for all: Is there any way (similar or not) to calculate the same Integral from 1 to infinity? It must be konvergent, because for X>2 the function x^-x < x^-2 so the Integral is smaller than the Integral of x^-2 wich is konvergent but how to calculate? I’m looking for this for decades…
I believe you must justify pulling the summation out of the integral. It is not enough to simply say that the sum (-1)^k/n! is a constant; k still appears in the integrand and these 2 operations do not always commute.
Perhaps rewrite this integral as being the integration from 0 to 1: x^(i^2 * x) dx
Maybe this might help. We can think of the exponent: i^2 * x as being periodic where x is the multiple or enumerated counts of a rotation by 180 degrees or PI radians.
If we look at i^2 as being sqrt(-1) * sqrt(-1) = -1. We know that the value of -1 is a 180-degree rotation of 1. We can use this along with the trigonometric properties to help solve this.
From what I can see, there is no one exact result from this definite integral. In other words, it's not going to give you a discrete area, volume or region under a curve or manifold. It appears that it is going to give you a series or sequence of them with a specific periodicity determined by the magnitude of x within the exponential component.
Thank you
What is the source for this solution?
You and someone else have the exact one, identical to the line, so I was curious if it is in a textbook somewhere.
That guy doesn't know the internet is for free.
Wouldn't your result mean that the area from 0 to 1 and from 1 to infinity are the same since alpha and x can be taken to be anything, even eachother?? Or am I wrong because you've introduced several changes of variables that places constraints on alpha?
Very nice presentation. Thanks.
Glad you liked it!
Great video
Very inspiring
this video on the other hand had no music. how outrageous. i can't believe you left out such a crucial element of a math video. because of lack of background music, i don't even know if i can call the video informative. i hope you will rethink your life choices that led you to choose to not put music over your video. literally unwatchable
edit: i skipped the intro and missed you saying there will be no music 😭 not sure if that makes things
I am happy there is no music (but no need to be so angry imho). Wonderfully explained but just a gentle constructive comment is maybe try to write a little more clearly. I know what is going on if I listen along but if I am just looking at the integral I start thinking what is planck's constant popping up in inappropriate place. Haha - just kidding - but you wrote the k really well once so I know you can do it :)
Yes this is a common theme, I'll wrote more clearly from now on 😅
There's a strange vagueness to the question here. How do you know that that answer is the final answer and can't be 'simplified' further? What functions would be allowed in that 'simplification'?
I didn't knew we could take out summation out of integrals.....are there conditions for this?
Absolutely, there are conditions, but if it all works out in the end those conditions probably held.
I actually liked the music, it jus doesn't have to be loud
Don’t complain, don’t explain. Just keep to the topic.
Good video, but it could use some music. Your soothing voice would work greatly with some jazz or something. Keep up the good content!
😂well done
3..
3?
@@Jagoalexander No, 3..
What was the actual question in the exam? Maybe I missed it but I don’t see the full question in the video or a link to it in the description. Was the question “Show (this integral) equals (this sum)”?
Yes it was, the actual question was show it's equal to the sum. Sorry for not explaining that well in the video. I was more focused on the method of solving the integral, the sum converges very quickly to around 1.2..
When I actually went to school, I was quite bothered by things that were distracting or incomplete, for a grade was going to come to hold me responsible for it just the same. Now, nobody is grading me on following it. Still, some others may be trying to keep up with school, and bringing their concerns with them, and perhaps a MITE irritated with the whole thing, because a grade is going to result at the end. I will not cuss you out, but I will urge you to maximize the helpfulness. k is k. It is not h, it is not u. And if on top of it, some random music is coming, this makes it harder. (Turning the sound off is not a good option, for you are speaking.) Thank you very much for understanding. It's not all about you either.
❤ awesome
Actually you didn't explain why you could interchange the integral and the sum. This is due to the dominated convergence theorem. We must explain all the details why we can do it
1:23 nice voic3 btw
Thank you :)
the way you write k can be interpreted as k, u or even h lol
I know, just my handwriting, I'll write more clearly next time!
my "n" looks exactly like his k
what is sum(0, inf)pow(alfa, -alfa) ? you're just gonna live it like that?
I suspect there is no closed form answer, but since we are providing feedback: a screenshot to wolframalpha, desmos or the like showing a numerical approximation is a great way to end these type problems.
Really interesting result but a laboured path through a lot of obvious steps.
Which steps are obvious - I wanted to provide a digestible solution to the average viewer who may not have advanced maths classes in their repertoire
Use Lambert W function maybe faster?🤔
very cool
I find it funny how the first step into finding the derivative and integral of x^-x are the same
Which software do you use for writing down your formulae?
Goodnotes on iPad
wouldnt you have to resubstitute in whatever alpha is to get the real result?
No, alpha is explicitly defined by the infinite sum's bounds
no music is fine thanks you !
What kind of sorcery is this
Praise be to God, mathematics is one of my best subjects. I got a score of 97 in the final exam, and the questions were very difficult. I am proud of my score because my students were crying because of the difficulty of the questions. I was hoping to get a score of 100, but I thank God for what I got. ❤
And we need to know this why? Is your ego sated now?
Putting music is fine as long as you are a musician and not talk over it :)
Changing the order of summation and integration can't be done carelessly, there should have been made some rigorous argument why this holds here.
It's called "proof by knowing that it can be done"
@@Jagoalexander ah, my favourite kind of proof
@@xenmaifirebringer552 it's like breaking a 4th wall, third time is a charm )
Yup that’s the only hard part, the rest is really easy
@@Jagoalexanderyou’re bad bro 😂
Nice analysis of the comment lmao
At any rate, your voice is already music to my ears
What answer did official Putnam authority give. Was it this one?
Yeah
@@Jagoalexander Ohh thanks!
All of this to write 2 times the same stuff
Hey, I see many comments about the handwriting, but it's completely understandable if you follow along, (I have terrible handwriting too).
👏👏😱👍
oh my god, fuck that hater
keep up the good work man
Thanks mate
I would say the problem solved in the video looks harder than the question asked in the actual exam.
Namely, asking: "find this integral " does not give you any clues. On the other hand, asking: "prove that this integral equals to this sum" is quite a clue. So the video is a bit misleading/clickbaiting.
Can you solve the integral from 1 to infinity of x^-x?
probably not. You can't swap the infinite sum for the integral if the upper limit of integration is at infinity
@@numbers93 Of cause, you can't solve this integral by infinite sum. But this integral definitely converge.(Because 0 < x^-x < e^-x when x > e, and the integral from e to infinity of e^-x converge) But I can't find any method to calculate the exact value of this integral.
That's one of the more bizarre comments I've ever seen.
It gave us a good laugh at least
Somehow youtube recommended this video to me and out of curiosity I opened it. Then I realized that this is the same person whom I learned Feynman Technique from lol! Subscribed!!
Thanks mate 👍 👌
So, a question I have is was the question on the Putnam in the form of “prove that \int_0^1 x^{-x}= \sum_{k = 1}^{\infty} k^{-k}”? Otherwise, I don’t think this would be a valid solution to the problem because it still isn’t in a constant form, it’s in the form of an infinite sum, which still needs different methods to solve/approximate. I took the Putnam twice, once in my freshman year of college, and again in my junior year, and unless it was worded like that, this would probably get a 1 or 4 or something like that, not a full 10
Yes, the question was show that the integral can be written as the infinite sum. The infinite sum converges very quickly to around 1.2… not an exact value unfortunately
9:00 why'd u keep saying k when writing n? or write n each time you say k?
geez writing is difficult (c) 3b1b
13:00 k
Both n and k are common variables to use with sums, however I've opted to use k as my variable I'm this video. In the video I do write the letter k but my handwriting makes it look like an n, making it confusing. But to clarify, I'm both saying and writing k
@@Jagoalexander it would be easier i guess to just say n ))
@@DeadJDonaBut then you'd be writing k and saying n!
@@_Heb_ how would you know that he's writing k then? )) or k factorial?
I could solve this integral in the bat of an eye. I chose not to 😎
4:50
Im afraid, Factorization of the constant at 6:26 is not correct
It is though