The Euler Mascheroni Constant
HTML-код
- Опубликовано: 14 окт 2024
- I define one of the most important constants in mathematics, the Euler-Mascheroni constant. It intuitively measures how far off the harmonic series 1 + 1/2 + ... + 1/n is from ln(n). In this video, I show that the constant must exist. It is an open problem to figure out if the constant is rational or irrational.
Application: Integral fractional part of 1/x from 0 to 1: • Integral of the fracti...
Monotone Sequence Theorem: • Monotone Sequence Theorem
Series playlist: • Series
Subscribe to my channel: / drpeyam
Check out my TikTok channel: / drpeyam
Follow me on Instagram: / peyamstagram
Follow me on Twitter: / drpeyam
Teespring merch: teespring.com/...
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
Oily-macaroni constant, served nice and hot.
Mayank KMC Papa flammy smiles upon you.
Standard pasta, always great, hot especially, twist then ingest.
YEAH served by our chef jens
@Mayank KMC oily-macaroni croissant 🥐
Missing some galois.
Never thought of that constant as difference of areas. This lecture is enlightening.
me too! :)
Euler only knew the 1st 16 digits by 1781. Mascheroni gave 32 in 1790 (with errors). Gauss 22 by 1811, Shanks 101 by 1871, Fischer and Zeller 1050 by 1961... Kim and Cutress over 600 billion as of May 2020.
Shanks ? One Piece reference?
@@lorenzosauditoyes
The fact that the difference between two divergent series converges to a constant is mind numbing.
I love the idea that beyond knowing some infinities are bigger than others we can in some cases find the difference between them! Cool video!
Agreed!!!
Thanks for your elegant explanation.
Dang I didn't know the proof that it converges was so easy! I always assumed it would be something super complex, but I totally could've done this one lol
14:21 Voiceover: *Can YOU figure it out?*
I like how you timed this with Mathologer's video. Great content :)
That convergency is beautifull, Euler was incredible matematician...
...is beautiful, Euler was an incredible mathematician
I love watching your videos.... totally inspiring
(14:15) haha! good one dr peyam. 🤣
1 = prob(γ is real) = prob(γ is rational or irrational) = [[ rationals and irrationals are disjoint ]] = prob(γ is rational) + prob(γ is irrational) = [[ rationals are countable ]] = 0 + prob(γ is irrational). Therefore γ is irrational with probability 1. 😁
Thank you Dr Peyam.
The sweetness in using Nonstandard and actually subtracting the infinities ;)
Mathologer, in his previous video said that the Ramanujan sum of 1/n (instead of being divergent) is the Euler-Mascheroni Constant which is impossible 😱😱😱
What makes you say it's impossible? Do you know what a Ramanujan sum is?
I remember discovering it myself when I was trying to sum reciprocals, and noticed the graph looked a lot like ln x, so I let my calculator run all night working out the difference that was about 0.577.
Not important, but I like to use ln(n+1) in the definition instead of ln(n). Then the regions match up better and you don't have the extra rectangle at the end.
Beautifully simple explanation. Thank you!
Dr. Peyam ( π-am) we often see the Euler Mascheroni Constant approximated as the harmonic number minus ln(n) minus 1/(2n). The 1/(2n), I noticed by using Maple, does make it converge much faster, but where does 1/(2n) come from?
How did you comment 2weeks ago in a 5 mins ago video
sumakit cjs reupload maybe? I don’t know.
Edit: It was probably unlisted until recently.
iirc that comes from the Euler-Maclaurin formula. -1/2 is the second Bernoulli number.
@@goodplacetostart9099 it must have been a reupload, though I wish I could time-travel.
It was unlisted
Amazing explanation
A more mathematical way of proving that Sn is bounded is transforming the sum into an integral using the floor function, then using the fact that the floor of x is smaller or equal to x, proving that the first integral is larger, and thus it is bounded above 0.
don't you mean roof function?
don't you mean roof function?
I love what he said "Not Neseccary since I love wasting peoples time." 🤣🤣🤣🤣🤣
Bless this man
Thanks for this video. Just fantastic.
I love this channel
Interesting and nice explanation...
Now here's the question, how do I approximate this using Minecraft?
Well, in Minecraft the difference between a block and a continuous function is zero, so...
probably something to do with armour stands
@@nullplan01 You're right, approximating it as a Riemann sum probably wouldn't work...
You could construct a computer in Minecraft, add a floating point unit and do all the calculations
@@aashsyed1277 I've done it in Minecraft! Have you not seen it?!?
A good approximation using mathematical constants and a physical constant:
gamma(0,577215665) < (PI/e)/2 = 0,5778636748954 gamma = (PI/e)/2 - alpha/((1+PI)*e) = 0,57721548319 alpha = Fine-structure constant = 0,0072973525643
Excellent presentation. Wow ! DrRahul Rohtak Haryana India
The Euler-Mascheroni constant is around 0.577 from what I've seen. My guess is that it is irrational. It's associated with natural logarithms, and that involves the natural base "e", which is transcendental, so how could the Euler-Mascheroni constant possibly be rational? My guess is that it's also not an algebraic number.
Sn+1 is smaller than Sn - agreed. How did you jump to Sn is greater than or equal to zero at 9.0. you have showed this geometrically later but can be rigorous. as usual very well made. please show the actual calculation of the value of the constant some time soon.
Wonderful explanation, very nice.
I finally understand this constant
Hi dr. Peyam, I am from Italy and I have the proof that the value of the volume integral of any f(x,y,z) within the region contained in your floor uplight is the same as the value of the integral of the same function evaluated within the region contained in mine...
Thanks for such a great explanation 🥰
it must to name of this only EULER his is the real legend but thank you for awsome axplining
Is there a practical closed formula for computing this constant? The only technique I can think of at the moment is to just directly compute S[n] by the definition (i.e, compute 1 + 1/2 + 1/3 ... + 1/n - Integral(1 ... n, (1/x)*dx)). Is there any known process better than this?
Sadly there is not
Great explanation
I have proved this firmness more easily
Very good video!
I like your tie!
Ah, good to be back in the Analytic classes :D
Really our thoughts make mathematics as a subject. BUT MATHEMATICIANS DOESN'T EVEN SEE MATH AS A SUBJECT IT IS THEIR LIFE 🗿🖤🖤🖤. SALUTE TO THE GOD~ euler.
dr peyam,
since you like analysis and making videos so much i think that you (and we as well) might enjoy a video (heck a series even) about a generalized mean or a power mean that (to my recent surprise and enjoyment) generalizes all means that we learn in school under one beautiful formula, namely
if -oo
That’s interesting, thank you!
How can the monotone theorem be true if it’s clear that the harmonic series itself violates this theorem (decreases and bounded but doesn’t converge)?
No, the series sum 1/n is not bounded, you’re confusing sequence with series
The spaghetti constant is superior
After proving it converges, the next logical step is to calculate the exact value it converges to.
Can you please do that?
That’s the thing! No one knows! The constant is gamma
@@drpeyam OK, not an exact value, but can you please show how to approximate it?
Well you just use the definition but with large n, like n = 1000. That should give you a better and better approximation since it converges
Dr. Peyam is a charming guy
Thankyou. One way to compare clocks.
Ok. Thanks.
Thank
Andile Mabaso a South African mathematician has published a PDF document online that he claims is a proof of the irrationality of the constant. and also proof that the constant is transcendental. Not being a mathematician, I am in no position to comment on the validity of his claim. As far as you know, Doctor Peyam, is Mabaso's claim legitimate? I have not seen any comment online about this, which strikes me as somewhat strange.
I doubt it
1:23 Your graph of 1/x is very inaccurate since the asymptote is the vertical axis, but your picture depicts the asymptote as being left of the vertical axis. It doesn't impact the video, but whatever.
Nope, the vertical axis drawn is x = 1 which is not an asymptote
@@drpeyam I didn't notice that. Thanks!
Thanks Dr 3.14159.....m
for a plate of Oily Macaroni ( constant ) .It was very tasty . DrRahul Rohtak Haryana India
Awesome!
Now ln(x) goes to infinity slowly. What series minus ln(ln(x)) converges to a constant?
Secondly exp(x) has a series where the difference between the series and the exp(x) converges to a constant???? And in case that is true what Riemann function are we talking?
1 + 1/2 + 1/3 + 1/4 + .... is only the Riemann function (sum of 1/(x^s), where s=-1). To me that is just a sloppy way of writing sum of x^s, where s can assume any real value......
exp x = e^x
Well, one way to generalize this is \sum^a_b f(x) - \int^a_b f(x). For e^x, we know it’s \sum^∞_{n=0} x^n / n!. So, using the generalization, the result would be:
lim x → ∞ \sum^∞_{n=0} x^n / n! - \int^∞_0 x^n / n! dn
Does it converge? Well, the sum converges to e^x, while integral converges to a non-elementary function. As for the difference, I don’t know. Plugging larger and larger x values into Wolfram Alpha seems to suggest that the difference converges to 0.
@@ruben3941 Wouldn't it be defined more like
limit as n->inf
[(sum from k=1 to k=n of 1/p_k ) - ln(ln(PrimePi(n)))] ?
Or are you saying you take limit as x->inf of the sum of the inverses of all primes p
S *can’t* assume and real value. It only converges for Re(s) > 1.
I will find it is rational or irrational
But my intuition say it is irrational
👍👍👍🌟🌟🌟🙏🙏🙏💯💯💯
Can you have a rectangle with base 1 and height 1?
Yes? A square is a rectangle
this constant always mesmerize me : there must be an other way to express Zeta(3) with ɣ than with this formula : ζ(3) = -1/2 ( ɣ³ + 1/2( ɣπ²) + Γ'''(1) )
Woowwwwwwww great one :O
That number is transcendental.
We don’t know
@@drpeyam eπ is irrational, along with e^e. None of them are known to be transcendental
@@WindowsXP_YT e*pi is not proven to be irrational.
What's Harmonic analysis
Adult Fourier Analysis
What's the value of that constant ?
Does it not have any specific value like pi or e?
Yes, it does. It's approximately 0,577.
Why don`t you ask Google?
"Does it not have any specific value like pi or e"
-_-
Martin Epstein Hahahahaha
Who can give me a summary about this constant &
Is anyone here going to celebrate gamma day on 5/7?
one over n plus one
Fun
γ
Nice italian pronunciation
hahahahah i was the first in here!
I was wondering who’d check out the video 😂
What the hell 6 months ago ?😐
I love this channel