The Most Intimidating Integral I've Ever Seen

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This Putnam Series was given on the Putnam exam in 1997 (Problem A3). Let's figure out a way to deal with this Putnam problem!
For those that don't know, the Putnam math competition features some super interesting and challenging problems!
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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.
Putnam 1997 A3
Putnam Exam 1997 A3
#math #brithemathguy #putnam

[@BriTheMathGuy]

🎓Become a Math Master With My Intro To Proofs Course! www.udemy.com/course/prove-it-like-a-mathematician/?referralCode=D4A14680C629BCC9D84C

[@blackpenredpen]

Beautiful presentation! Love it!

[@BriTheMathGuy]

Thank you so much!

[@joshuasusanto6626]

Wonderful! Simply wonderful! This I'm placing on my journal to sleep.

[@adritobiswas1982]

Yup

[@RisetotheEquation]

Outstanding. Sometimes I wonder who's more impressive: the student who solved the integral or the person who conjured it.

[@farukkaya4396]

Arms getting bigger, so is the channel!

[@BriTheMathGuy]

😂😅

[@flintsteel2700]

The echo is a little jarring but nonetheless still a beautiful solution to such an intimidating integral! Good stuff

[@sujitdey1717]

And i thought i was the one who felt something was different.

[@BriTheMathGuy]

Sorry about that! It should be fixed in the future.

[@sujitdey1717]

@@BriTheMathGuy no problem the math was great as always. Love you and your content. 💙.

[@yohangross5518]

That's a very very beautiful way of solving a particularly intimidating integral, you just won a suscriber

[@BriTheMathGuy]

Thanks so much!

[@victorhernandez-eg7wp]

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.

[@BriTheMathGuy]

Wonderful!

[@MrThrashMan98]

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.

[@BriTheMathGuy]

We all crave it! Thanks for watching after your tough day!

[@lucas0m0james]

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)

[@archieharrodine3925]

That feeling when n factorial cancels

[@yousifkhalil9655]

Watched to the end, liked, saved to favorite math playlist, already subscribed, there isn't just anything left to do.

[@shivam5105]

Become a Putnam fellow

[@BriTheMathGuy]

Wow thank you so much!

[@aashsyed1277]

@@BriTheMathGuy me too!!!!!!!!!!!!

[@thegodoflols]

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

[@SaidVSMath]

Your videos kick ass man, I want to make ones just like them! I love this fast paced but concise format

[@BriTheMathGuy]

Thanks so much! Best of luck!!

[@rishavchoudhuri8806]

Subscribed!! Brilliant way of solving the integral as well as presenting it. Loved the video!

[@BriTheMathGuy]

Awesome, thank you!

[@manucitomx]

Wow! What a great way with words! I love your channel.

[@BriTheMathGuy]

Thanks so much! Have a great day!

[@eduardoduque6288]

What a tremendous exposition! New subscriber! Thank you for your material! 🌹🔥

[@tmsniper9229]

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

[@kummer45]

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.

[@maxkempf7687]

That is a suprisingly beautiful result! Thank you for covering this in a video. :D

[@BriTheMathGuy]

My pleasure!

[@PunmasterSTP]

This was amazing; thank you so much for sharing!

[@BriTheMathGuy]

Glad you enjoyed it!

[@yusufdenli9363]

That was amazing👏👏 Congratulations

[@BriTheMathGuy]

Thanks so much!!

[@yomonsbuzz4978]

i appreciate this hope that maths will be fun and famous like nothing before once

[@diegopablogordillovaras106]

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!!

[@BriTheMathGuy]

Many thanks!

[@tomkerruish2982]

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.

[@angelmendez-rivera351]

It would be helpful if you are familiar with the Bessel functions, since they satisfy many integral equations.

[@yoav613]

Your videos are so fun to watch.😃

[@BriTheMathGuy]

Glad you like them!

[@sicapanjesis3987]

Your presentation of the solution always gets me. My best wishes to you and please please continue

[@BriTheMathGuy]

Thank you, I will!

[@kafianan6586]

The way you explain, makes these intimidating integrals seem easier

[@BriTheMathGuy]

Glad you think so! Have a great day!

[@yahav897]

I really enjoy these videos! Can't wait to start taking higher level maths in uni

[@BriTheMathGuy]

I'm so glad!

[@ianbryant]

I like that you get into the math immediately

[@lizs004]

Such a great video!

[@BriTheMathGuy]

Glad you liked it!!

[@kinshuksinghania4289]

Wow!! Absolutely marvelous!!

[@BriTheMathGuy]

Thank you! Cheers!

[@mehex9858]

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*

[@BriTheMathGuy]

😂

[@blazedinfernape886]

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!?!?!

[@Rzko]

no

[@nikhilnagaria2672]

yes

[@10-year-oldcalculus19]

Yesn’t

[@georgeryandev2103]

Yes, making mistakes is good for the growth of math skills.

[@rotsovanvorleak4704]

Such the one of the best teacher ever

[@djridoo]

So good video !

[@BriTheMathGuy]

Thanks so much!

[@aloneworm7]

4:15 When he said "we still have some exes lingering about' , I felt that

[@purim_sakamoto]

うおおおお　Bravo!! めっちゃくちゃわかりやすかったです!!!😍😍😍👍👍👍

[@BriTheMathGuy]

Thanks so much!

[@ashishraje5712]

Great u make maths lucid

[@majdsaleh_]

You release that you're good at math when u start watching the contents in x2

[@ArjunBhanap]

Incredible explanation!

[@BriTheMathGuy]

Glad you think so!

[@marble17]

2:18 My dirty brain just hears a curse word

[@dharrshanmahadevan9450]

i have no clue what he is talking about but i still love it

[@romanoemmanuelle1967]

Beautiful!

[@BriTheMathGuy]

Glad you enjoyed it!

[@maximevanderbeken4712]

Nice proof ! Now you just need to justify swapping the sum and the integral.. as it cannot always be done .

[@santiago_moralesduarte]

The sum converges to less than e^(u/2)

[@frankreashore]

Very nice. Thanks.

[@BriTheMathGuy]

Most welcome!

[@arnavsoni1701]

You really should become a math professor....

[@BriTheMathGuy]

Currently an instructor (no Phd)😅

[@arnavsoni1701]

@@BriTheMathGuy great!!!

[@scotttidwell262]

Great video, thanks Bri. What program are you using for the text?

[@loshwoof311]

cool integral, great video:D

[@BriTheMathGuy]

Glad you liked it!

[@mohammedal-haddad2652]

Incredible!

[@BriTheMathGuy]

Glad you thought so!

[@adithyar4282]

bro pls make more videos on putnam integrals .They are really interesting. Thank you in advance

[@muse0622]

I can explain this integral just one word. WOW

[@BriTheMathGuy]

🤯

[@axbs4863]

That worked out so perfectly lmao

[@user-wy3mr6nj6w]

Amazing!

[@BriTheMathGuy]

You are!

[@tueur2squall973]

5:05 why can we do this ? Permute the sum and the integral? Is it because the sum is converging uniformly on [0,+infinity] ?

[@BrollyyLSSJ]

I'd say dominated convergence theorem, with something like exp(-u+u/2) = exp(-u/2) being the integrable dominant.

[@Rzko]

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.

[@tueur2squall973]

@@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

[@tueur2squall973]

@@Rzko And yeah Thank you , I did understand the following steps

[@Rzko]

@@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)

[@deekshanaik2438]

I lost track for the first few times but I'm glad I understood this in the end :)

[@BriTheMathGuy]

it's a tricky one! :) thanks for watching!

[@deekshanaik2438]

@@BriTheMathGuy yea your vids are quite interesting... Who knew a bio nerd like me would binge math questions some day... Thanks for ur efforts

[@hydropage2855]

This is awesome

[@kangmoabel]

This video is sooooo Good ❤❤ from Ethiopia, Africa

[@EssentialsOfMath]

Nice, I was able to do this one! Really awesome integral

[@BriTheMathGuy]

Great job!

[@nqa1893]

Always awesome like you are :-)

[@BriTheMathGuy]

You're the best!

[@malexmartinez4007]

The awkward moment when a solution is as pretty as the one presenting it.

[@holyshit922]

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

[@normanfrancisco2063]

Woah!!! Mind blown...

[@BriTheMathGuy]

🤯

[@minecraftherobrine1234]

This is so good

[@BriTheMathGuy]

Glad you thought so!

[@aayushsinghsengar7129]

this goes way beyond advanced level of that pesky JEE

[@dfsfssdfsdfs3084]

Really? I thought the advanced JEE was the hardest test

[@manojsurya1005]

🤩, it was a crazy integral, it involved power series gamma u sub,I want more integrals like this

[@endormaster2315]

Incredible

[@BriTheMathGuy]

Glad you thought so!

[@atrumluminarium]

That was quite an aesthetic one

[@viniciusmoretti]

You say that to all of them.

[@a.osethkin55]

Super!

[@BriTheMathGuy]

Thank you! Cheers!

[@aakashkhamaru9403]

Amazing.

[@BriTheMathGuy]

Thank you! Cheers!

[@tommoritz1007]

That was so enjoyable

[@pamir8232]

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.

[@violintegral]

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

[@braedenlarson9122]

I love watching *other* people do integrals :)

[@Diaming787]

Is there a Taylor series expansion that expands to the factor of (n!)^2?

[@felipegomabrockmann2740]

that was pretty cool

[@BriTheMathGuy]

Glad you thought so!

[@henryginn7490]

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

[@kummer45]

Use double factorials. These are useful.

[@azurev2258]

it's crazy how something that looks absolutely nasty like this can simplify down into √e at the end

[@Observer_detector]

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

[@sujalsalgarkar360]

These vedios are really good but I want to a one on some mathematical concepts or theory.

[@BriTheMathGuy]

I'll do my best!

[@ashikak5743]

Intimidating ❤️

[@BriTheMathGuy]

😬

[@jakewenczkowski1056]

Awesome

[@quantumgaming9180]

Mathematics always blows up my mind

[@BriTheMathGuy]

🤯

[@iamtrash288]

Does the second series converge on 0 to INF though? I can't really see it because I suck at evaluating them functional series.

[@umylten4142]

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.

[@puceno]

at 3:30 u had a chance to turn that sum into e^2x*sum(n=0,infinity,1/2^2n)

[@aashsyed1277]

You did a HARD putnum problem in 6 minutes! So impressed ! I think you are a genius !

[@magicmulder]

He explained the solution in 6 minutes. No telling how long it took him to find the solution.

[@aashsyed1277]

@@magicmulder no but at least he is a genius .........

[@magicmulder]

@@aashsyed1277 He is very good, but most math students could solve that one. Genius is rare. Very rare.

[@PixelSergey]

This one was very beautiful

[@BriTheMathGuy]

Glad you thought so!

[@polychromaa]

Are there any other places where i could find an integral like this with two sums multiplied together in the integrand?

[@ARBB1]

Nice answer

[@BriTheMathGuy]

Glad you thought so!

[@spsubjectparaphrased8377]

Simplified results are beauty gives extraterrestrial vibes.

[@daphenomenalz4100]

Why do we always end up at gamma in these types of problems 😂,

[@BriTheMathGuy]

I don't know! 😂

[@alienbroccoli8296]

Satisfying Answer

[@BriTheMathGuy]

I think so too!

[@raunak1147]

I got the first sum. But was clueless about what to do with (n!)^2... Subbing x^2/2=u was brilliant bruh

[@ShinySwalot]

It all folds together!

[@BriTheMathGuy]

Right?!

[@AcademyOmen]

Nice🍀

[@BriTheMathGuy]

Thanks ✌

[@insouciantFox]

This is really dangerous from a rigour perspective. The 2nd power series leads to dominated convergence issues that most will just hand wave away, but they are there.

[@aaryanjogina]

I liked this

[@fasihullisan3066]

awesome

[@victorribera5796]

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

[@God-ld6ll]

integradating

[@BriTheMathGuy]

😂