what does gamma have to do with it?

[@sarithasaritha.t.r147]

Sorry babe not now, michael penn just dropped.

[@gnarlybonesful]

slight error at 12:55, ln(3n) = ln(3)+ln(n) so the ln(n) should be positive

[@brahimsebbata9036]

Sur....but hé consulate it

[@radonato]

That's about the only thing I understood....

[@dzuchun]

integral producing oily macaroni

[@SAHarry]

Love the calc videos, they really help me to explore beyond the normal first year uni stuff.

[@oliverwalser284]

agreed

[@iWillWakeYouUp]

This one hit the spot. It's enjoyable to see a true professional work on problems like this and make it presentable for people like me who don't study mathematics, but know just enough to follow. He's also gotten much more considerate of people like me, who probably make up a considerable portion of his audience. Thanks for explaining stuff, Michael. I enjoy it when you re-explain some stuff, put up theorems, or go back to a previous line on the board and explain where something came from. The only thing that trips me up is whenever there are typos in these videos, cause then I try to follow but can't.

[@skylardeslypere9909]

I never expected it to have such a clean solution at the end

[@averagegamer9513]

6:40 I think you meant the finishing term is x^(3n-3). The Penn Fact shown is wrong, the sum should either run up to n-1 or it should be the sum from 1 to n of ar^(k-1).

[@krisbrandenberger544]

The sum should run up to n-1.

[@Anonymous-zp4hb]

Lovely result. This one had me on the edge of my seat.

[@robertoxmusica]

Fun fact: if you take a random number x between 0 and 1, the expected value of the decimal part of 1/x is also 1-gamma

[@skalderman]

One would expect 0 which is 1:0.5=2.0 so why is there a skew?

[@Slackow]

@@skalderman there's a difference between taking the average of a function and plugging it into another, and composing the two functions into one function, and taking the average of that. Edit: to try to make this more clear let E be the expected value function: E(f(g(x))) ≠ f(E(g(x)))

[@NathanSimonGottemer]

This is technically an improper integral because the sum in the integrand diverges to infinity at the upper bound of integration, but thankfully our buddy Ramanujan helps us out here (and also helps explain how one of Euler’s constants showed up lol) Edit: fixed typo. Stupid autocorrect.

[@curtiswfranks]

That was enjoyable. I saw where it was heading, but it was really satisfying to see that conclusion. That is a pretty weird result!

[@egillandersson1780]

Amazing ! A little bit tricky but amazing. Thank you !

[@bennyloodts5497]

Beautiful adventure!

[@daviddragas149]

When we define the integral in terms of I of n 13:33 how can we set I of 3^n to be the equal to it when we have another 3 in the exponent on x which means in the exponent we get 3^(n+1). Shouldnt the integral then equal to I of 3^(n-1)

[@Mystery_Biscuits]

Very nice surprise appearance of the Euler-Mascheroni constant! I'll be honest, I didn't see that coming until the introduction of J_n

[@sergiysidenko]

This final sum goes from n=1 to infinity, from 0

[@erikstanton3908]

Shouldn't it be x^3^n not x^3n?

[@moonwatcher2001]

Awesome!!!

[@tolberthobson2610]

Cool, MORE CRAZY INTEGRALS!!!!!!

[@pierreabbat6157]

Is it x^3n or x^3^n?

[@ere4t4t4rrrrr4]

is the sum over x^(3^n) or x^3n?

[@beniborukhov9436]

Seems like it starts with the first and then switches to the second. This was driving me a little insane. If you change the second term in the original infinite sum to x^6 and ignore the x^(3^n) on the board it works out.

[@edwardlulofs444]

I had a lot to learn in this episode.

[@mantisbog]

Question: What happens if you add Kurt Angle to the mix?

[@alexdemoura9972]

I couldn't find the link to the video about Harmonic numbers he refers to at 18:14 - could somebody post it as a reply here. Thanks.

[@MichaelPennMath]

here you go: kzbin.info/www/bejne/kJrdq4qef8eepNE -Stephanie MP Editor

[@alexdemoura9972]

@@MichaelPennMath thanks

[@nathanmorlonski1265]

how can I self study math for olympiads

[@thetheoreticalnerd7662]

Correct me if I'm wrong, but the 17:30 seems a bit hand wavy; we need to first show that the sequence containing this subsequence converges, and then we deduce that it converges to the same limit.

[@schweinmachtbree1013]

Well he's taking the limit itself as "well-known" so it's fine - that is, we're recognising H_{3^{N+1}} - ln(3^{N+1}) as a subsequence of a (known) _convergent_ sequence, not just as a subsequence of a sequence

[@thetheoreticalnerd7662]

@@schweinmachtbree1013 Perhaps though it would have been better to mention this before writing out the limit, but I digress.

[@schweinmachtbree1013]

@@MyOneFiftiethOfADollar People have repeatedly told you that you are extremely rude for posting this kind of comment and yet you continue to do so. May I ask if you are on the spectrum?

[@thetheoreticalnerd7662]

@@schweinmachtbree1013 thank you for this

[@iWillWakeYouUp]

6:25 Could someone explain to me why the sum goes to n-1 instead of n?

[@lucasdepetris5896]

in the penn fact, it is actually up to n-1, not up to n

[@iWillWakeYouUp]

@@lucasdepetris5896 I see, just a typo then. Thank you.

[@sushildevkota350]

@@iWillWakeYouUp not that. Actually he starts from 0 so upto n-1. You can write 1 to n. Its upto you.

[@iWillWakeYouUp]

@@sushildevkota350 Ah ok, my bad then. Thank you!

[@iWillWakeYouUp]

@@sushildevkota350 So it was a typo after all.

[@jeffreyrodrigoecheverria2613]

You man an error, you meant to write x^(3n) and not x^3^n which is something else.

[@michaelzumpano7318]

Oh, that was a great ending. Of course gamma! So the polynomial normalizes the hypergeometric integral?

[@neilmorrone691]

Derivations, Integration, Summations have always been manageable for me. Now your Examples of combinations of MATHEMATICAL Tools have re-invigorated my attention to the World of MATHEMATICS. Grazie! I'm considering acquiring your Lessons, and/ or the "BRILLIANT" treatment of various Subjects.

[@birdbeakbeardneck3617]

that's hyper in the name. am scred lol

[@veselindimov307]

Great video, Prof. Penn, really enjoyed this problem. Wonderful editing as always, Stephanie! Around 11:30 I thought that it is enough to sum up I_1+I_3+I_9+I_27+... When I do this, the harmonic numbers cancel each other out (as Prof. Penn showed in the end) but I am left with an infinite amount of ln3-s which will sum up to infinity. Am I doing sth wrong/taking for granted/missing?

[@bjornfeuerbacher5514]

Not all harmonic numbers cancel. You are left with something like H_1 + lim_N to infinity ( ln(3^(N+1)) - H_(3^(N+1)) ), and the limit just gives minus the Euler-Mascheroni constant.

[@brahimsebbata9036]

Best ..i remembre that for 24 years ago

[@williamsullivan857]

Longtime viewer and really appreciate the work that you do here. So to the reason I'm posting: How do we know that we have convergence?

[@jmgraves8]

The geometric series converges so long as |x^3|

[@pdc4458]

This is erroneous, x^3^n is not x^3*n. After that i stopped following.

[@MyOneFiftiethOfADollar]

And we are all soooo devastated. Can this channel survive without your “astute error checking”? Why waste time commenting here when you could making big bucks tutoring?

[@BridgeBum]

He didn't say that they were. He doesn't immediately give a reason why he is using x^3n in his "lemma", but it comes together.

[@MyOneFiftiethOfADollar]

@@BridgeBum right, some viewers expect Dr Penn to fill in every last detail, but that would be too time consuming.

[@yoav613]

@@MyOneFiftiethOfADollar yes,he already explains in detail,but there are always chronic complainers.

[@MarcusCactus]

Hello, Mr Penn. I enjoy your videos very much. May I mention that, in my opinion, the new thumbnail style is too flashy, attracting more the moronic types than the intended viewers, while it should better convey the rationality of the content. Such as it is, it makes one think it is an ad for a HYPERmarket or something, I almost threw it in the garbage bin. Thank you anyway for the math.

[@lgooch]

Integrating the hyper geometric function?