The Euler-Mascheroni Constant - An Epic Proof of Convergence!

[@jorgesponja3042]

"Negative times negative becomes positive I did a proof on that" Our boi Jens, who proofs our finest dreams

[@newkid9807]

Flammable Maths fuck You

[@gustavosedano294]

Wheler macarroni?

[@ffggddss]

Oily macaroni. Fred

[@MrRyanroberson1]

I prefer the geometric proof. it's quite simple: Graph both functions, 1/x and 1/floor(x). ignoring the region between 0 and 1, look at the difference. the functions always meet whenever x is an integer, and each partial difference has its own exclusive y range. Therefore each difference can be moved to be vertically aligned, and they all fit in a unit square.

[@willyou2199]

"gamma = 0.7" ~papa flemmy 2k19

[@neilgerace355]

By the Fundamental Theorem of Engineering, gamma = 1/sqrt(2)

[@davidrheault7896]

@@neilgerace355 gamma is roughly 1/sqrt(3)

[@oni8337]

@@davidrheault7896 engineer

[@SteamPunkLV]

I haven't watched your videos in a while, but now that I finally have some time, I decided to check them out. I had completely forgotten how amazing this channel is

[@Soepler]

it can be shown that this is the constant term of the taylor/laurent expansion of the riemann zetafunction at 1 btw summation by parts would have done the job in 2 lines:)

[@kaychimav]

Can you elaborate? Sounds promising

[@faresmarseille3972]

@@kaychimav what he is saying is that as x approches 1, zeta(x) can be approximated to 1/(x-1) + gamma i.e. that the limit as x approches 1 of zeta(x) - 1/(x-1) = gamma

[@jkid1134]

This is a brilliant video, really stunningly simply put. And on my birthday no less! Thank you so much for all the work and prep work videos that went into this; I’ve never felt more close to the macaroni number.

[@emredogan2102]

Did you just say APPROXIMATION laughs in engineer:

[@funkysagancat3295]

First vídeo that I see here that is actually really easy to understand

[@G_sen_sei]

The sequence has a lower bound , monotonically decreasing. It's a nice blackboard! I want to put it in my house. その黒板メッチャいいですね。

[@pedrolourenco9520]

Okay this is epic

[@OnamKingtheKing]

Wheeler macaroni

[@hoodedR]

Hi Papa... Greetings from Munich!

[@andiback]

small correction at 17:48 gamma is about 0.5772, btw you can speed up the calculation of gamma by adding 0.5 within the ln() expression (second convergence order) therefor the formula reads gamma = lim n->inf (H(n) - ln(n+0.5))

[@neilgerace355]

15:50 Ma Ma Ma Mascherona! With apologies to The Clash

[@shacharh5470]

There's no relation between The Clash and this song, you must have meant another band

[@neilgerace355]

@@shacharh5470 You are right! The Knack, not The Clash :)

[@cubicardi8011]

-lim- *liー*

[@neilgerace355]

Li Li Li Li Li Li - Simon and Garfunkel, "The Boxer"

[@dexter2392]

li... *some weird arabic script*

[@hamsterdam1942]

-ln- L

[@eladnic]

I very much enjoyed this video!

[@eliasarguello9961]

Can you please do a video on the divergence of the sum of the reciprocals of the primes, another video on the convergence of the difference of the sum of the reciprocals of the primes and ln(ln(n)) to the Meissel-Mertens constant, and maybe another video on an integral representation of the Meissel-Mertens constant? They would be fun videos, especially as a follow-up to the recent videos you posted! I can't wait to see them :)

[@eliasarguello9961]

Papa Flammy im dumb and accidentally fixed something in my comment so can you please heart it again? Thanks ❤️

[@sarthakvarshney7784]

Great video once again! Just a minor addition. In your Taylor expansion of ln(1+x), I believe you didn't add the condition that -1

[@angelmendez-rivera351]

Sarthak Varshney The condition is not necessary since it was stated that n is a natural number, and as such, 1/n necessarily satisfies 0 < 1/n =< 1.

[@nishatmunshi4672]

@@angelmendez-rivera351 was worried about it the whole time. Thanks for explaining

[@mohammedal-haddad2652]

Beautiful argument. Thank you very much.

[@avtaras]

0:15 Lol the gamma speaks for itself

[@farkarf]

The way I remember this argument (I think I first saw it on Mathologer) is to remember Dr Flammable's argument for it having a lower bound. That lower bound argument is a comparison of the harmonic series with the integral of 1/x. Here it was established by comparing histograms of unit width with the integral of 1/x over that same interval, much as in introductions to calculus. If the unit width histograms begin (and end) at integer values, then taking the height of each histogram to be the value of 1/x at the end of each interval, one establishes it is bounded from below (as in the video above); conversely, if the height of each unit-width histogram is taken to be the value of 1/x at the *beginning of the interval, then an upper bound can be established. So I guess my point is the construction for the last argument already cuts both ways: for both being bounded from above, and from below.

[@vermeerdragyl341]

No Nut November. That's not even a problem. Not Math November? HOLY SH*T, I surrender!

[@M.Davit613]

I have proved this firmness more easily

[@maximunnit]

you should use a capital sigma as a summation variable

[@rome8726]

I asked for this question months ago and I finally get the answer😂😂. Thanks Papa.

[@goncalofreitas2094]

Very good Papa! Thanks so much for the Wheler-macarroni constant!

[@goncalofreitas2094]

@@PapaFlammy69 :)

[@user_2793]

First time I watched your video in an official capacity. They randomly threw Euler Mjcnwncwn constant at us in diff eqns

[@a0z9]

Ha sido épico. Mas épico será que alguien relacione esta constante con otra constante llamada razón aurea. Ambas ponen en relación los números primos consecutivos de una forma bien concreta que ayudará a desvelar la demostración más épica sobre los números primos.

[@emilpysenisoncrack420]

1:22 Favorite part of the whole video

[@weerman44]

Such a beautiful proof! Thanks Jens!

[@AG-hl7bg]

Oh man I love this kind of stuff.

[@shrimatkapoor2200]

Most satisfying video evvvahhh

[@PapaFlammy69]

:33

[@bhaveshgohel9249]

Good proof!

[@tetraedri_1834]

Correction: gamma_n > 1/n does not give strict bound gamma > 0, as gamma could still be zero. Sanity check: 2/n > 1/n, hence in the limit 0>0. ;)

[@HasteCS]

I’m addicted to these videos and I have no idea what the fuck is going on

[@11kravitzn]

A somewhat simpler way: gamma_n=H_n-ln(n+1) (n+1 vs. n doesn't matter in the limit) gamma_n=sum from 1 to n [1/k-ln(1+1/k)] As f(x)=x-ln(1+x) is the integral of t/(1+t) from 0 to x, clearly 0

With your argument you showed that gamma is greater or equal to 0, not strictly greater than 0, because the limit of a strictly positive sequence may be 0.

[@brunocaf8656]

Apparently, you can also derive an identity involving gamma and the zeta function, by expanding that ln(1+1/n) in taylor series, obtaining: gamma = SUMn=2:infty (-1)^n zeta(n) / n

[@PapaFlammy69]

Already made a video on that! :)

[@user-kr6bp4zi2y]

This guy has the Integerahld-pass

[@user-kr6bp4zi2y]

aber ohne scheiß, als du gesagt hast "sigma balls" hätt ich beinahe meinen Tee verschluckt

[@thephysicistcuber175]

That summation variable though

[@ozzyfromspace]

Papa Flammy: “Good morning my fellow mathematicians” Me: *yawning* how did you know it’s 4:51am EST?

[@hakimben8054]

your videos are amazing !

[@hakimben8054]

@@PapaFlammy69 you are welcome .

[@RandomDays906]

You also get gamma from taking the limit as h->0 1/2 (zeta(1+h) + zeta(1-h)).

[@doodelay]

Here's the thing, and I haven't watched the proof yet, but when I just saw this equation pop up in my textbook I got kind of annoyed because infinity - infinity is "SUPPOSED" to be indeterminate. Yet, here we are, taking the difference of two divergent series and (i guess) getting a definite answer. UPDATE: Nvm, I see the constant is found differently than I imagined :o

[@PapaFlammy69]

:)

[@douglasstrother6584]

The Euler-Mascheroni Constant is one crazy number! γ = ∞ - a little smaller ∞ ! mathworld.wolfram.com/Euler-MascheroniConstant.html

[@SuperBCRich]

you blew my speaker

[@maxpercer7119]

nice to see your'e enjoying math, bringing positive energy to a difficult subject ;o

[@justdusty9697]

i want that shirt

[@AlgyCuber]

the oil in macaroni constant

[@PapaFlammy69]

:'D

[@filippodifranco8225]

Since you made a video on how to pronounce Deutch scientists names you know how I feel each time you say Masceroni instead of Mascheroni ;-) In Italian ch is equivalent to k as pronounce and it is used together e and i only. So the right pronounce is like it was written Maskeroni.

[@فارسالزعبي-ف3د]

Thank you very much

[@vukstojiljkovic7181]

Love this video!

[@Chefiscool]

this was fucking brilliant

[@monamahdi4972]

Great job....thanks

[@andrealosardo5973]

Great video, but pls 4 the italian viewers like me: Mascheroni has an italian pronunciation like "Maskeroni"

[@livedandletdie]

But Fappable Maths is German so Mascheroni is ma sch eron i.

[@hacker2ish]

When you did the integral definition and broke up the integral into smaller parts I think the best way to continue the proof is to say that each integral from a to a+1 is less than (a+1 - a) * 1/a becausw 1/a is the largest value on the interval and thus proving that the series is always positive and thus bounded by 0 from below

[@timka3244]

You deserving 1000000000000000000000 subs

[@hacker2ish]

9:13 right side of the chalkboard looks like an eye

[@jameswilson8270]

Very good video. By the way, the theorem you used is called the Monotone Convergence Theorem (also called the Bounded Monotone Convergence Theorem). I'm not sure why you didn't state that in the video. But anyway, I see what you did there with that trivial statement. You used a nice little trick. All you needed to do, of course, was to note that 1/sigma was greater than or equal to the integrand over all values in the interval, and, therefore, the integral of 1/sigma over the interval is greater than the integral of 1/t. I enjoyed that clever trick though. It shows off the versatility of mathematics arguments.

[@MuPrimeMath]

Drinking game: take a shot every time Papa says "I'm terribly sorry"

[@rot6015]

Woow I really liked this proof

[@rot6015]

normie ass comment

[@dgrandlapinblanc]

Et voilà ! Thanks.

[@Jack_Callcott_AU]

Some interesting facts about the harmonic numbers H_n. Obviously they are always rational, and they increase without bound as n tends to infinity, but no harmonic number, except the first, is an integer...surprising, is it not.

[@immersionmusic]

Int & GRAAL

[@216betto]

Oily macaroni!!

[@arandomghost8819]

I like oily macaroni !

[@bondarakhanganeh]

Nice Gimmel

[@helloitsme7553]

So if I understood correctly, the proof that γ_n+1>γ_n proved that the sequence decreased for all n, and then the other proofs prove that γ_n is bounded between 0 and 1, thus only by the first statement the sequence can either diverge to -∞ or decrease to a constant but the second statement makes sure it does not diverge to -∞ cause that is not between 0 and 1?

[@matteodamiano6733]

Yes

[@electric7487]

Euler Mascheroni constant? More like Oily Macaroni constant!

[@pedrolourenco9520]

What's this constant useful for tho? I've never seen any application of it so far

[@FlashBack2B]

Actually it appears virtualy everywhere in real analysis. I'd even say that it is the third most important constant of real analysis, below pi and e. It appears for instance in the Weierstrass representation of the Gamma function, in the laurent series of the Riemann Zeta function, and in countless integrals, series and limits

[@russellbarnes6111]

FlashBack2B yeah it’s crazy how often you see it. I want to say it shows up in both the lower bound for the totient function as well as the lower bound of the gap between primes. It shows up in just about everything with prime numbers and I don’t really know why. Probably some connection to the zeta function

[@willyou2199]

Its the principle value of the diverging harmonic series. in essence its like the simplest kind in the "sort of infinite yet finite number" group. That's why it appears in fields where divergences have to be handed, like riemann zeta, feynman integrals

[@pedrolourenco9520]

Today I was able to use this constant to prove a certain area was divergent, the constant technically got cancelled out, but because of its definition I was able to rewrite harmonics series in terms of ln which made everything easier

[@caesarinchina]

why not use that integral with reciprocals of x, one with the floor function? just interested in seeing how it works actually, seems elegant

[@JoseWui]

Corporate wants you to find the difference between these two photos.

[@PraneshPyaraShrestha]

Oily Macaroni

[@PlayandGameTV]

I got really seri(es) ligma from watching this video

[@hoodedR]

Is it enough to prove that a limit is bounded to prove that it exists?? sin x is bounded between [-1,+1] but obviously l(squiggles)x->inf sinx doesn't exist

[@PapaFlammy69]

Nah, being bounded it not enough! (squiggle xD that made me lel :D)

[@cecilhenry9908]

Is the only way to calculate EM just brute force???

[@esdrasmunizmota8933]

You can prof this using Just the unequality: exp(x)>=x+1 for all real valeu of x.

[@peterjohnson4854]

When the constant is a meme itself...

[@peterjohnson4854]

Papa noticed me. I feel like a convergent boi

[@atrimandal4324]

Oh my god :O

[@soumaknandi9218]

YOU'RE NOT GONNA FIND THE MACARONI VALUE YOURSELF?! smh my head

[@leonalmeida2411]

I'm doing a school project, I would like to know where did you find this proof (or if you come up with it)

[@avdrago7170]

17:49 Gamma is actually around 0.577

[@dannygjk]

Easy way to remember the decimal value (roughly) is tan 30 degrees :)

[@travorliu1192]

Hi Papa, would u like to create a video that proves the convergence of Stieltjes constants

[@matkach200]

Can you do some Quantum Chromodynamics?

[@otonanoC]

Lorenzo Mascheroni was Italian. There is no "sh" sound in his name. It is MASS-KAIR-RONI.

[@FitR_MusicProductions]

Ay y’all ever heard of that wheeler-macaroni constant?

[@priyanshubansal6776]

at 11:09 what he said i didn't get . what language he was using ?

[@MathEnthusiast-od8yu]

I think to show that the sequence is bounded above are unnecessary bec ause the sequence are decreasing

[@RetroGamingClashOfClans]

cant you show the derivative is approaching zero as input increases so it just means its approaching to a stop at some value of convergence...

[@gamma_dablam]

Derivative approaching zero doesn't imply convergence to a finite limit. Case in point the natural log function

[@jksmusicstudio1439]

lel, this was on my calc 2 exam QQ

[@jarahfluxman20]

Jens: Lauer bound... ajf besser als ich in Deutsch, der "wachsen" und "wichsen" verwechselte.

[@shambosaha9727]

17:51 Did you prove that γ is an irrational number? Because nobody ever has managed to prove it, as far as I know.

[@PapaFlammy69]

nope

[@zoltankurti]

The lebesgue measure of Q is 0. So if you pick a random number it's irrational. Gamma has a stupid random definition, so gamma itself is random. Gamma is irrational.

[@shambosaha9727]

@@zoltankurti You should totally publish this and get a Fields Medal or somethin'.

[@zoltankurti]

@@shambosaha9727 I don't have time for that. Right now I'm checking all the non trivial zeros of the zeta function to infinite decimal digits precision by hand to finally be done with this dumb riddle people keep talking about.

[@Gamma_Digamma]

Euoli Maschener

[@bandamkaromi]

Frankly speaking, I could not trace this topic in subway. watch again. :'(

[@edwardlatt7383]

did not prove the limit.. and the EM constand is not .7.. it's .577... 0.57721566490153286060651209008240243104215933593992.. to be exact

[@nuklearboysymbiote]

Oily macaroni

[@lional1010]

Thanks alot for do my favorite line as clear 😚😚😚😚 but there other thing is on board please please zoom it using pubg mobile scope 😂😂😂😂😂 yes thanks alot again

[@Somestupiedbudee]

Yooler-Mascarpone

[@WindowsXP_YT]

γ

[@danielkirilov8065]

Gucci