7:20 i think this 0.4 step thing is due to the term t^n. Basically that term can work for negative values of t when n (rational) as a reduced fraction has an odd denominator, because otherwise you would be taking the sqrt of a negative number. It works for multiples of 0.4=2/5 because 5 is odd. Then, the left branch appears over the x axis because the 2 at the numerator squares the negative number. So I suppose this phenomenon appears also with numbers such as 2/3, 4/3, 6/7 etc...
The cusps you get in the function come from the graph of the real valued function y=x^a (when a is a fraction p/q where p is even and q is odd). Just try this in Desmos with a slider for a. E.g. y=x^(2/3), y=x^0.8.
Also, I think the integral converges everywhere on the set C\(N\{0}), meaning that so long as the real part is not a negative integer, it should converge. I'm not 100% certain, but it should follow from the functional equation itself.
7:30 d/dt (t^1.2) = 1.2*t^0.2 and if you take the real branch of the 5th root this is continuous at t=0 (all other components are differentiable as well). This would also work, e.g., with n=4/3.
@@PapaFlammy69 why in hel would anyone ever think to bring an e function into this..surely there is a way tomsolce without this at all..just with algebra or something?
It looks like it approximates a Gaussian for larger n. Have you considered the Laplace method to obtain an asymptotic? I know that it works as a method of obtaining the standard Stirling approximation
Interesting stuff, I wonder if you kept summing up each function what would happen to that exponential integral form that was reached. If you were to start with numbers, factorials, K(n), sum(K(k)), sum(sum(K(k))), ... etc whether there would be a pattern in how you evaluate the integral. o_o
What does analytic continuation mean here? Could you please expedite its contents? And also what was that function you used ( K ) i couldn't follow that!! Thank you for your brilliant support!!
If you don't want to derive the identity in 5-10 videos, you could always just make a movie :P I think more people would be likely to watch it that way too.
Can u plssss do a video on what kind of educational courses you took as a math major, and why did u take maths etc and how does the education system work in Germany
Bruh I'm watching this video at work because I wanted to find out how to find the sum of factorials and the first video suggestion on google is the Pappa
I tried to find the algebraic formula. What a fool I was! In grade 9 my math teacher gave me a problem: Σn!:3^a=N; find highest value of a when n=2013, except, for ya know, it's were formulated quite differently. At this point I already knew that if there are some laws of physics, there must be formula for me calculating it by hand on paper, so there must be formula for the sum of factorials. And it found to be that monstrosity. What a troll she was.
I'm no math whiz so every time I see people interchange the integral and summation notation it grinds my gears. Anyone willing to provide a link for easy understanding?
Rishabh Vailaya There is nothing to worry about, since this summation was finite. Therefore, you can interchange them because integration is linear, so the integral of the sum must be the sum of the integrals. It would be a different story if you had an infinite sum, which we did not have here. For infinite sums, you have to worry about uniform convergence in the interval of integration, and then you apply the dominated convergence theorem.
math.stackexchange.com/questions/83721/when-can-a-sum-and-integral-be-interchanged Hopefully this link is of some help, I don't think it's an intense or difficult read and it's fairly concise.
Wait.. for the dominated convergence theorem you don't need the uniform convergence right? U need the pointwise convergence of the function series, the continuity and the domination by an integrable function iirc
Watching your videos only makes me hungry for more knowledge. Man, Imma be honest with you here. Feels like you get turned on more and more every time you get closer the the answer. Ngl, I feel the same. 😆
![Factorial's Asymptotic Expansion - DERIVING STIRLING'S FORMULA! [ +Desmos Insights ]](http://i.ytimg.com/vi/-k7s_nXGuO0/mqdefault.jpg)
When there’s no poll…
sad times :(((
oh mann i have looked for this everywhere and i find it here.yay
:))
@Oily Macaroni i tried to find it on youtube since it is easier to understand. also i didnt look that hard it was just one or two searches
I think I will stay with adding my factorials by hand (or calculator), looks easier then this integral
:D
I feel brilliant watching this.
:DDD
Time to flex my lack of mathematical rigour when this clock arrives.
:^))) Thank you for the support! :D
Die Gamma! Try Gamma! Why Gamma! Oh My Gamma!
:D
Gamma gamma down dooby doo down down...
7:20 i think this 0.4 step thing is due to the term t^n. Basically that term can work for negative values of t when n (rational) as a reduced fraction has an odd denominator, because otherwise you would be taking the sqrt of a negative number. It works for multiples of 0.4=2/5 because 5 is odd. Then, the left branch appears over the x axis because the 2 at the numerator squares the negative number. So I suppose this phenomenon appears also with numbers such as 2/3, 4/3, 6/7 etc...
Build 'em big!
Just the thing for the first day of Summer.
yas! :D
The cusps you get in the function come from the graph of the real valued function y=x^a (when a is a fraction p/q where p is even and q is odd). Just try this in Desmos with a slider for a. E.g. y=x^(2/3), y=x^0.8.
Where do I find more about the engineering clock? I can't find it in the description
Look at the pinned comment or my story please! It'll be available mid next week, you will see an update in your subscriptions then! =)
The listing is finally live btw! :D teespring.com/flammable-math-clock
Also, I think the integral converges everywhere on the set C\(N\{0}), meaning that so long as the real part is not a negative integer, it should converge. I'm not 100% certain, but it should follow from the functional equation itself.
yas, I believe so too Angel! =)
When you said it was a sexy integral, I felt that on another level.
7:30 d/dt (t^1.2) = 1.2*t^0.2 and if you take the real branch of the 5th root this is continuous at t=0 (all other components are differentiable as well). This would also work, e.g., with n=4/3.
thx! =)
But papa what is the representation for the alternating sum of k! ?
Good question! I'll look into it! :D
Anyone else feel weird when he talked about how hard the graph explodes
:^)
I actually needed this, but didn't know what to search for online. Thank you German man.
Playing with desmos graphs has honestly been a big part of my math education.
:)
Very nice one! Also that integrll sign was hot af
You could do a video about the analytic continuation of tetration to the real numbers, it would be pretty amazing!
Blessings for Brilliant for sponsoring you
:)
You're so Passionate with Mathematics!
:)
What is the name of the program that was used in the video to show the graphs?
Desmos :)
Thanks thanks thanks
in geogebra, the numerator had the power n instead of n-1, did I miss something here
I just indexed it differently, nothing special happening^^
I don't know how hard it is but I would love to see the derivation of the definition with the exponential integral. Would an hour special do it papa?
don't think so honestly ^^'
@@PapaFlammy69 why in hel would anyone ever think to bring an e function into this..surely there is a way tomsolce without this at all..just with algebra or something?
I have no clue what this is but it looks cool
:D
Tbh I'd watch the derivation of the wacc boi you mentioned. Was feeling nostalgic for Coxeter's integral the other day; might be the same gucci sht
budget Zach Starr demonstrates that factorial thing diverges
xD
Which tool do use to plot the graph?
Desmos. It's free and online if that helps
I got recommended and I'm not disappointed
=D
Oh ho, this looks like a build up to something absolutely magnificient!
:))
would it possible to assign a divergent value to the sum as n approaches infinity?
probably ^^
The sound in 7:03 made me check if I unmuted my whatsapp web tab lol
:'D
Pretty sweet functions, didn't expect it to be that small.
:)
It looks like it approximates a Gaussian for larger n. Have you considered the Laplace method to obtain an asymptotic? I know that it works as a method of obtaining the standard Stirling approximation
The way you pronounce Kurepa is hilarious 🤣. nic vid btw
thx :p
Is there a partial sum formulated for 1/n²?
Try to use integrals too! :)
@@PapaFlammy69 ok, i try it
Interesting stuff, I wonder if you kept summing up each function what would happen to that exponential integral form that was reached. If you were to start with numbers, factorials, K(n), sum(K(k)), sum(sum(K(k))), ... etc whether there would be a pattern in how you evaluate the integral. o_o
bruh
Wow this is the first time I came soo early..🙌🙌😁
m2
That's what she said
Good stuff my boy! A nice little nugget indeed. Can't wait to see what you have in store pour moi ;)
:vvv
Yo that’s a nice looking answer. Good job Flammy ma boi
:3
What does analytic continuation mean here? Could you please expedite its contents? And also what was that function you used ( K ) i couldn't follow that!! Thank you for your brilliant support!!
in layman's terms: expanding the radius of convergence
Kurepa's function =)
Can you make a Video where you solve the integral from 0 to infinity from 1/x! dx?
Already done
Flammable Maths nice. Thanks a lot and greetings from Frankfurt!
Thank you for making this video.
no problem Stefan :3
Thanks and good job bro.
If you don't want to derive the identity in 5-10 videos, you could always just make a movie :P I think more people would be likely to watch it that way too.
:)
i feel like that meme is a personal attack on me
:D
You can do integrals in Desmos you known, even graph them.
Amazing clock!!!
Soon available! I'll keep you informed! =)
The listing is finally live btw! :D teespring.com/flammable-math-clock
Very interesting video... i really enjoyed it and found it interesting 😀
=)
You can justify the use of noninteger n in the integral representation as a generalization of the discrete sum via fractional finite sums.
I have done sigma k*k! Which amazed me but you will really give me a 💓 attack
How does a factorial graph look like pappa?
Look at the gamma function wolfram documentation! =) It's pretty wild! :D
Can u plssss do a video on what kind of educational courses you took as a math major, and why did u take maths etc and how does the education system work in Germany
Already did that! Watch my Grade review :)
I have an amazing result, papa flammable maths. The sum of the natural numbers up to n is n+n^2/2.False or True?
yas
@@PapaFlammy69 i don´t remember integrals because ahh im not a math guy.
Bruh I'm watching this video at work because I wanted to find out how to find the sum of factorials and the first video suggestion on google is the Pappa
i cant deal with how you write your sigmas. hahaha But fantastic video and great work as always
:D
I tried to find the algebraic formula. What a fool I was! In grade 9 my math teacher gave me a problem:
Σn!:3^a=N; find highest value of a when n=2013, except, for ya know, it's were formulated quite differently. At this point I already knew that if there are some laws of physics, there must be formula for me calculating it by hand on paper, so there must be formula for the sum of factorials. And it found to be that monstrosity.
What a troll she was.
Hey you
Yes you papa
why do you not have discord server?
I do
@@PapaFlammy69 then where's the link D:
Papa’s turning into Sex star ??!!!wait till he finds out 😂😂😂
you make my brain down 😁😁
hehe :D
Yes, definitely understood some words :p
:D
I'm no math whiz so every time I see people interchange the integral and summation notation it grinds my gears. Anyone willing to provide a link for easy understanding?
I was just making use of the fact that the integral is a linear operator ^^
Rishabh Vailaya There is nothing to worry about, since this summation was finite. Therefore, you can interchange them because integration is linear, so the integral of the sum must be the sum of the integrals.
It would be a different story if you had an infinite sum, which we did not have here. For infinite sums, you have to worry about uniform convergence in the interval of integration, and then you apply the dominated convergence theorem.
math.stackexchange.com/questions/83721/when-can-a-sum-and-integral-be-interchanged
Hopefully this link is of some help, I don't think it's an intense or difficult read and it's fairly concise.
@@angelmendez-rivera351 thanks! I'm actually an engineering student so I'm not familiar with everything the channel covers :)
Wait.. for the dominated convergence theorem you don't need the uniform convergence right?
U need the pointwise convergence of the function series, the continuity and the domination by an integrable function iirc
Integral of Gamma function!!! (factorial factorial factorial)! (factorial)! (fakturiul)
Where tf is the polls!??
YT got rid of the festure :(((
@@PapaFlammy69 Fuck, y?
What’s with the high pitch voice?
yeyeye :v
Before watching the video, i know the answer wont be 1/e, but my heart says it should be 1/e
*k+1
yeye
Don't you think ...you need to improve your handwriting !
Anyways, it's awesome 👌 love it .
Wait k! isn't gosper summable...
Why the long video.
It's obviously 1/e
rib
No, it's -1/12
@@user_2793 bruh
Took him less than a minute to say analytical.
yeye
Is fractional factorial possible
What is fractional factorial for 8400000
???
Wow you look awful lot like gla1ve from Astralis
#subscribed
I spend half an hour trying to find a solution to the what was apparantly the final result :(
xD
I DIDNT NITICE WHERE DID POLLS GO!!!????
Removed by YT :(
Watching your videos only makes me hungry for more knowledge.
Man, Imma be honest with you here. Feels like you get turned on more and more every time you get closer the the answer. Ngl, I feel the same. 😆
:D
I would like to submit my proof of Rieman's Hypothesis:
This is an elementary fact and has been left as an exercise for the reader.
Unfunny and irrelevant.
You are doing a sin, buoy # Kratos. Pi²=g, wtf!! 😂😂
La serie geometrica t^k è valida per |t|
woww nicee
I got distracted by your t-shirt..
:'D
Great
$18.69.. I see what you did there 😫
:^)
Koooreeepaaa
Anyone give a number and I will tell the sigma of factorial in seconds
Bruh i was watching some minecraft vids, how did i get here..
xD
The polls 😢
:((((
Your gamma looks like a sigma haha
Maybe your sigma just looks like a gamma? :^)
Will papa notice me if I comment this early ?
He will notice you no matter at what time you comment
ye :p
Great job! Andrews Mom ~ 💕
Thank you! :)
recommention to find first meme =)
Everything good, till I noticed pi^2=g.
xD
Die Katze schmeckt gut
jawoll!
Hello
osu :vvv
so you didn't talk about your gucci snake, now you are gonna take its venom....😜🐱🐉🐱🐉🐱🐉🐱🐉🐱🐉its neck snake
say my name.
hey Arnab boi :v
wow Are we connected by a sacred Khala Or are you reading my mind? XD XD I'm so glad I subscribed to your channel.
:D
Noice
Chill down with the sex stuff bro, its not that interesting comming from you