That was so insightful. I have never dealt with an integral like that, but now I am confident that if I ever see one, not to panic. Thank you! I really enjoyed this video.
Watching these videos makes me realize that my hunger for scientific knowledge is still stronger and bigger than my fatigue after a full-time, warehouse-assistant working day.
Very nice presentation! To be absolutely rigorous though, it'd be nice to mention that each of the series converge for all positive x (ratio test) and that the sum and integral can be interchanged (e.g. tonelli's theorem)
The integral of the sum is the sum of the integrals because the integral is a linear function. Then you just put out of the integral the terms that don't have 'u', which means they are constants.
@@Rzko U can do this when Everything is finite , I mean when the sum is finite , but when It's a series (infinite sum) , you need more argument : you need to know if the sum is converging , how it's converging in order to switch the sum with the integral
5:11 i think swapping the integral and the infinite sum there requires using the dominated convergence theorem(if we think about it rigorously), very good presentation overall
Phenomenal!! Your way of presenting a problem is mind-blowing. Discussing the possible methods in a step, how to start solving it, best approach ... Everything illustrates how good you are in math and throws light on the beauty of math
The second power series (1 + x²/2² + ...) equals the the Bessel function of the first kind J_0 evaluated at ix, although I don't know how that would be helpful in this problem.
Beautiful. I am so proud of myself that I solved it on my own. Edit: Okay maybe I didn't solve it completely correct lol I messed up a 2^r and got the answer e instead of sqrt(e)........ that is fine right!?!?!
Beautiful problem, and very beautiful answer. Using the sum representation of the exponential function and the Gamma function… what a ride haha. Love your channel!!
For the first factor I did the following pulled out x, substituted u=-x^2/2 For the second factor i have got second order linear differential equation but not with constant coefficients xy''+y'-xy=0 Second factor will probably be Bessel function but when we get first factor Gamma function will be helpful
The second part can also be written as (x^n)^2 / ((2^n)^2 * (n!)^2) and we can take the entire term into square like (x^n / 2^n * n!) ^2 which we can write as ((x^n/2^n)/n!)^2 = ((x/2)^n /n!) ^2 so we can put it into e^x form like (e^(x/2))^2 which basically is e^x.
I hear so much stuff about the Putnam being ridiculously hard, but every step here was the most obvious thing to do given the current stage. Like it's not something you just scribble down in a hurry, but it's something I imagine most mathematically experienced people could do. Lovely presentation though
can you cancel the x, in the same quick step where you cancel the 2^n?? since one of the limits of integration is 0 you would have a 0/0 in x, I'm not sure if it is allowed to cancel out the x there
The second series absolutely converges. If I call U(n) = {x^(2n)}/{[2^(2n)]*(n!)^2}, then you have: U(n+1)/U(n) = {x^2}/{4(n+1)^2}, which converges to 0 as n -> infinity, for all x in (0, infinity). By the ratio test, that series converges.
3:35 HOW CAN THIS APPEAR ON MY RECOMMENDED AFTER I HAVE JUST MISTAKEN EXACTLY THAT THING AT THE TUESDAY TEST!??!?!?! THAT EXACT du=x*dx IS where u=(x^2)/2 IS THE EXACT THING I MISSED! I didn't realise x can be written as the derivative of (x^2)/2 at the test and i magically passed it even with that mistake.
I like your videos very much. One tiny suggestion though- can you slow down your speed while explaining such problems. You go very fast, which is problematic to understand what you are saying. I mean, before even I understand the concept you told, you move to another concept.
Hi I graduated from mathematics in Science in 2000 from Iraq and I have been in Australia 11 years.I have recognised in my Bachelors degree from Australia and I did in Australia certificate 3and4 in Accounting 9 years ago and I didn’t work all past years because of some circumstances I had related to my family . Now I am thinking for work and I wish something related to my math bachelor degree instead Tafe certification for better job even I can start from step 1 but in the future it will leads me to good job in the math field . I know you are not in Australia but I would like to ask you if you could advise me what kind of job I have to find related to what I have with no experience and from long time of graduation and also in case if there’s an any support courses related to the gap made after graduation to be suitable or competent for work .Sorry I am asking you but I am thinking you might help me from where I can began . I was thinking for doing master in math but I found it well be harder to me now after 21 years of the graduation even I got about 85% in bachelor in math all university years or excellent in Accounting (but for only one year) . Thank you in advance. Rana
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Beautiful presentation! Love it!
Thank you so much!
Wonderful! Simply wonderful!
This I'm placing on my journal to sleep.
Yup
Outstanding. Sometimes I wonder who's more impressive: the student who solved the integral or the person who conjured it.
Definitely the professor
Arms getting bigger, so is the channel!
😂😅
That was so insightful. I have never dealt with an integral like that, but now I am confident that if I ever see one, not to panic. Thank you! I really enjoyed this video.
Wonderful!
Watched to the end, liked, saved to favorite math playlist, already subscribed, there isn't just anything left to do.
Become a Putnam fellow
Wow thank you so much!
@@BriTheMathGuy me too!!!!!!!!!!!!
That's a very very beautiful way of solving a particularly intimidating integral, you just won a suscriber
Thanks so much!
The echo is a little jarring but nonetheless still a beautiful solution to such an intimidating integral! Good stuff
And i thought i was the one who felt something was different.
Sorry about that! It should be fixed in the future.
@@BriTheMathGuy no problem the math was great as always. Love you and your content. 💙.
Watching these videos makes me realize that my hunger for scientific knowledge is still stronger and bigger than my fatigue after a full-time, warehouse-assistant working day.
We all crave it! Thanks for watching after your tough day!
Very nice presentation! To be absolutely rigorous though, it'd be nice to mention that each of the series converge for all positive x (ratio test) and that the sum and integral can be interchanged (e.g. tonelli's theorem)
5:05 why can we do this ? Permute the sum and the integral?
Is it because the sum is converging uniformly on [0,+infinity] ?
I'd say dominated convergence theorem, with something like exp(-u+u/2) = exp(-u/2) being the integrable dominant.
The integral of the sum is the sum of the integrals because the integral is a linear function. Then you just put out of the integral the terms that don't have 'u', which means they are constants.
@@Rzko U can do this when Everything is finite , I mean when the sum is finite , but when It's a series (infinite sum) , you need more argument : you need to know if the sum is converging , how it's converging in order to switch the sum with the integral
@@Rzko And yeah Thank you , I did understand the following steps
@@tueur2squall973 are you sure about that? An infinite sum is just the limit of a partial sum (idk if we say like that in english)
That feeling when n factorial cancels
5:11 i think swapping the integral and the infinite sum there requires using the dominated convergence theorem(if we think about it rigorously), very good presentation overall
That was amazing👏👏 Congratulations
Thanks so much!!
Your videos kick ass man, I want to make ones just like them! I love this fast paced but concise format
Thanks so much! Best of luck!!
Wow!
What a great way with words! I love your channel.
Thanks so much! Have a great day!
The way you explain, makes these intimidating integrals seem easier
Glad you think so! Have a great day!
The format of black screen, the math in all the details and the clean process with all the steps makes these series of tutorial useful.
Phenomenal!!
Your way of presenting a problem is mind-blowing.
Discussing the possible methods in a step, how to start solving it, best approach ...
Everything illustrates how good you are in math and throws light on the beauty of math
The second power series (1 + x²/2² + ...) equals the the Bessel function of the first kind J_0 evaluated at ix, although I don't know how that would be helpful in this problem.
It would be helpful if you are familiar with the Bessel functions, since they satisfy many integral equations.
By looking at that integral, I instantly understood that I would not be able to solve it if I try.
*And I was not disappointed*
😂
Subscribed!! Brilliant way of solving the integral as well as presenting it. Loved the video!
Awesome, thank you!
Your videos are so fun to watch.😃
Glad you like them!
I can explain this integral just one word.
WOW
🤯
You did a HARD putnum problem in 6 minutes!
So impressed !
I think you are a genius !
He explained the solution in 6 minutes. No telling how long it took him to find the solution.
@@magicmulder no but at least he is a genius .........
@@aashsyed1277 He is very good, but most math students could solve that one. Genius is rare. Very rare.
That is a suprisingly beautiful result! Thank you for covering this in a video. :D
My pleasure!
4:15 When he said "we still have some exes lingering about' , I felt that
Beautiful. I am so proud of myself that I solved it on my own.
Edit: Okay maybe I didn't solve it completely correct lol I messed up a 2^r and got the answer e instead of sqrt(e)........ that is fine right!?!?!
no
yes
Yesn’t
Yes, making mistakes is good for the growth of math skills.
What a tremendous exposition! New subscriber! Thank you for your material! 🌹🔥
Beautiful problem, and very beautiful answer.
Using the sum representation of the exponential function and the Gamma function… what a ride haha.
Love your channel!!
Many thanks!
For the first factor I did the following
pulled out x,
substituted u=-x^2/2
For the second factor i have got second order linear differential equation but not with constant coefficients
xy''+y'-xy=0
Second factor will probably be Bessel function
but when we get first factor Gamma function will be helpful
You really should become a math professor....
Currently an instructor (no Phd)😅
@@BriTheMathGuy great!!!
2:18
My dirty brain just hears a curse word
i have no clue what he is talking about but i still love it
this goes way beyond advanced level of that pesky JEE
Really? I thought the advanced JEE was the hardest test
I like that you get into the math immediately
The awkward moment when a solution is as pretty as the one presenting it.
うおおおお Bravo!!
めっちゃくちゃわかりやすかったです!!!😍😍😍👍👍👍
Thanks so much!
You release that you're good at math when u start watching the contents in x2
Your presentation of the solution always gets me. My best wishes to you and please please continue
Thank you, I will!
i appreciate this hope that maths will be fun and famous like nothing before once
Nice proof ! Now you just need to justify swapping the sum and the integral.. as it cannot always be done .
The sum converges to less than e^(u/2)
Why can't it always be done? In this case isn't it just a constant in the integral? Still learning so I'm genuinely curious
Such the one of the best teacher ever
So good video !
Thanks so much!
This was amazing; thank you so much for sharing!
Glad you enjoyed it!
Wow!! Absolutely marvelous!!
Thank you! Cheers!
at 3:30 u had a chance to turn that sum into e^2x*sum(n=0,infinity,1/2^2n)
I really enjoy these videos! Can't wait to start taking higher level maths in uni
I'm so glad!
🤩, it was a crazy integral, it involved power series gamma u sub,I want more integrals like this
hmmmm.. that integral can simplify like
ʃ (1-x*exp(-x^2))*BesselI(0,x) dx
and BesselI(0,x) is modified Bessel Function of the First kind
I love watching *other* people do integrals :)
The second part can also be written as (x^n)^2 / ((2^n)^2 * (n!)^2) and we can take the entire term into square like (x^n / 2^n * n!) ^2 which we can write as ((x^n/2^n)/n!)^2 = ((x/2)^n /n!) ^2 so we can put it into e^x form like (e^(x/2))^2 which basically is e^x.
You made a mistake. In general, given a sequence a_n, the sum of (a_n)^2 is not equal to (the sum of a_n)^2
Is there a Taylor series expansion that expands to the factor of (n!)^2?
Such a great video!
Glad you liked it!!
I hear so much stuff about the Putnam being ridiculously hard, but every step here was the most obvious thing to do given the current stage. Like it's not something you just scribble down in a hurry, but it's something I imagine most mathematically experienced people could do. Lovely presentation though
That worked out so perfectly lmao
Incredible explanation!
Glad you think so!
1:21, you can't use the same summation variable for the two sums, 2nd one should be 'm', or whatever, but not 'n'.
Wrong, those aren't nested sums, it's a product of sums, meaning the variable names do not share the same scope and therefore cannot collide.
Great u make maths lucid
That was quite an aesthetic one
I lost track for the first few times but I'm glad I understood this in the end :)
it's a tricky one! :) thanks for watching!
@@BriTheMathGuy yea your vids are quite interesting... Who knew a bio nerd like me would binge math questions some day... Thanks for ur efforts
This one was very beautiful
Glad you thought so!
Thank you. You have tought me that I really am (x^2)/2
This video is sooooo Good ❤❤
from Ethiopia, Africa
This is awesome
You say that to all of them.
Mathematics always blows up my mind
🤯
Maybe you should make math memes review that would be fun!
I'll see what I can do!
bro pls make more videos on putnam integrals .They are really interesting. Thank you in advance
why echo?
Sorry about that! It should be fixed in the future.
can you cancel the x, in the same quick step where you cancel the 2^n?? since one of the limits of integration is 0 you would have a 0/0 in x, I'm not sure if it is allowed to cancel out the x there
Use double factorials. These are useful.
cool integral, great video:D
Glad you liked it!
Always awesome like you are :-)
You're the best!
Beautiful!
Glad you enjoyed it!
it's crazy how something that looks absolutely nasty like this can simplify down into √e at the end
What's with the audio issues?
Sorry about that! It should be fixed in the future.
I got the first sum. But was clueless about what to do with (n!)^2...
Subbing x^2/2=u was brilliant bruh
This is so good
Glad you thought so!
Simplified results are beauty gives extraterrestrial vibes.
Does the second series converge on 0 to INF though? I can't really see it because I suck at evaluating them functional series.
The second series absolutely converges.
If I call U(n) = {x^(2n)}/{[2^(2n)]*(n!)^2}, then you have:
U(n+1)/U(n) = {x^2}/{4(n+1)^2}, which converges to 0 as n -> infinity, for all x in (0, infinity).
By the ratio test, that series converges.
Is the echo on purpose ?
Had an issue while recording, Sorry about that! It should be fixed in the future.
Are there any other places where i could find an integral like this with two sums multiplied together in the integrand?
3:35 HOW CAN THIS APPEAR ON MY RECOMMENDED AFTER I HAVE JUST MISTAKEN EXACTLY THAT THING AT THE TUESDAY TEST!??!?!?!
THAT EXACT du=x*dx IS where u=(x^2)/2 IS THE EXACT THING I MISSED! I didn't realise x can be written as the derivative of (x^2)/2 at the test and i magically passed it even with that mistake.
Great video, thanks Bri. What program are you using for the text?
I like your videos very much. One tiny suggestion though- can you slow down your speed while explaining such problems. You go very fast, which is problematic to understand what you are saying. I mean, before even I understand the concept you told, you move to another concept.
Amazing!
You are!
Nice, I was able to do this one! Really awesome integral
Great job!
Intimidating ❤️
😬
how do you have the cool summation stuff?great vid btw
Very nice. Thanks.
Most welcome!
Why do we always end up at gamma in these types of problems 😂,
I don't know! 😂
It all folds together!
Right?!
These vedios are really good but I want to a one on some mathematical concepts or theory.
I'll do my best!
Satisfying Answer
I think so too!
Woah!!! Mind blown...
🤯
Incredible!
Glad you thought so!
Engineer: 👁️👄👁️
Physicist: It looks like a Taylor series so It must be something related to e^x. And IT cant be a big power
Mathematician:
Hi I graduated from mathematics in Science in 2000 from Iraq and I have been in Australia 11 years.I have recognised in my Bachelors degree from Australia and I did in Australia certificate 3and4 in Accounting 9 years ago and I didn’t work all past years because of some circumstances I had related to my family . Now I am thinking for work and I wish something related to my math bachelor degree instead Tafe certification for better job even I can start from step 1 but in the future it will leads me to good job in the math field . I know you are not in Australia but I would like to ask you if you could advise me what kind of job I have to find related to what I have with no experience and from long time of graduation and also in case if there’s an any support courses related to the gap made after graduation to be suitable or competent for work .Sorry I am asking you but I am thinking you might help me from where I can began . I was thinking for doing master in math but I found it well be harder to me now after 21 years of the graduation even I got about 85% in bachelor in math all university years or excellent in Accounting (but for only one year) . Thank you in advance. Rana
You are so awesome 😎😎😎😎!
You're litterally everywhere! 🤣🤣
@@tl1989 helooo
@@aashsyed1277 Hiiii!
You are!!