Infinite Series - Numberphile

Рет қаралды 435,254

5 жыл бұрын

Fields Medallist Charlie Fefferman talks about some classic infinite series.
More links & stuff in full description below ↓↓↓
Charles Fefferman at Princeton: www.math.princeton.edu/people...
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Пікірлер: 805

@JJ-kl7eq 5 жыл бұрын

Introducing the Numberphile video channel which absolutely will never, ever be discontinued - The Infinite Series.

@b3z3jm3nny 5 жыл бұрын

RIP the PBS KZfaq channel of the same name :(

@JJ-kl7eq 5 жыл бұрын

Exactly - that was one of my favorite channels.

@michaelnovak9412 5 жыл бұрын

What happened to PBS Infinite Series is truly a tragedy. It was my favorite channel on KZfaq honestly.

@-Kerstin 5 жыл бұрын

PBS Infinite Series being discontinued wasn't much of a loss if you ask me.

@johanrichter2695 5 жыл бұрын

@@-Kerstin Why? Did you find anything wrong with it?

@snowgw2 5 жыл бұрын

Hello, you can't end it like that. Not without explaining how it becomes Pi^2/6

@4dragons632 5 жыл бұрын

I agree completely. I really want to know as well. But a quick wikipedia dive suggests that this topic would take at least a whole video of it's own. I hope they're going to do it because I'm getting equal parts confused and fascinated by this.

@RedBar3D 5 жыл бұрын

Agreed! Let's hope they follow it up with another video.

@ipassedtheturingtest1396 5 жыл бұрын

My professor did the same thing in our calculus script. Just wrote "actually, you can show that this series converges to π²/6." and left it there. Might be a great strategy to encourage curious students (or viewers, in this case) to think about it for themselves, though.

@sirdiealot7805 5 жыл бұрын

He also fails to make an argument for why he thinks that the first series ends up as equal to 2.

@andretimpa 5 жыл бұрын

The easiest rigorous proof iirc involves finding the Fourier Series of x^2, so it would take a bit more of explaning. You can look up "Basel Problem" in wikipedia for more info

@Kilroyan 5 жыл бұрын

can I just compliment the animations in this video? in terms of presentation, numberphiles has come such a long way, and I love it!

@tablechums4627 2 жыл бұрын

Props to the animator.

@lazertroll702 2 жыл бұрын

I miss the days of simple shorn parchment and sharpie.. 😔

@erumaayuuki 5 жыл бұрын

Matt Parker used this series and equation to calculate pi on pi-day with multiple copies of his book named Humble Pi.

@incription 5 жыл бұрын

of course he did, haha

@frederf3227 5 жыл бұрын

Ah yes I remember how he got 3.4115926...

@Danilego 5 жыл бұрын

@Perplexion Dangerman wait what

@InDstructR 5 жыл бұрын

@@frederf3227 yes... 3.411....

@brennonstevens467 5 жыл бұрын

@Perplexion Dangerman ~arrogance~

@ilyrm89 5 жыл бұрын

My mind cannot handle the different kind of paper!

@debayanbanerjee 5 жыл бұрын

Yep. Stands out like a sore thumb.

@rebmcr 5 жыл бұрын

It seems they had a shortage of brown paper rolls and decided to use brown envelopes instead!

@BloodSprite-tan 5 жыл бұрын

for some reason they are called manilla envelopes, i suggest you check your eyes, because that color is not brown, it's closer to a buff or yellowish gold.

@lucashermann7262 5 жыл бұрын

Its okay to be autistic

@rebmcr 5 жыл бұрын

@@BloodSprite-tan well it's a lot flipping closer to brown than white!

@zuzusuperfly8363 5 жыл бұрын

Shout out to whoever did the work of adding the animation of an enormous sum that only stays on screen for about 2 seconds. You're the hero. Or depending on how it was edited, the person who wrote it out. Edit: And the person doing the 3D animations.

@pmcpartlan 5 жыл бұрын

Glad it's appreciated! Thanks

@ruhrohraggy1313 5 жыл бұрын

An infinite number of mathematicians enter a bar. The first one orders one beer, the second one orders half a beer, the third orders a quarter of a beer, the fourth orders an eighth of a beer, and so on. After taking orders for a while, the bartender sighs exasperatedly, says, "You guys need to know your limits," and pours two beers for the whole group.

@Oskar5707 11 ай бұрын

I'm stealing this😎

@bo-dg3bh 10 ай бұрын

lol poor mathematicians

@CCarrMcMahon 5 жыл бұрын

"PI creeps in where you would least expect it..." and so does this video.

@Triantalex 8 ай бұрын

false.

@maxpeeters8688 5 жыл бұрын

Another fun bit of mathematics related to this topic: In the video, it is explained that 1 + (1/2) + (1/3) + ... diverges and that 1 + (1/2)^2 + (1/3)^2 + ... converges. So for a value s, somewhere between 1 and 2, you could expect there to be a turning point such that 1 + (1/2)^s + (1/3)^s + ... switches from being divergent to being convergent. This turning point happens to be s = 1. That means that for any value of s greater than 1, the series converges. Therefore, even something like 1 + (1/2)^1.001 + (1/3)^1.001 + ... converges.

@samharper5881 5 жыл бұрын

Yes. Any infinite sum of (1/x)^a is Zeta(a) (the Riemann Zeta function and there's a video of it on Numberphile) and zeta(>1) is always positive. Zeta(1.001) (aka Zeta(1+1/1000)) as per your example is a little over 1000 (1000.577...) Zeta(1+1/c) as c tends to infinity is c+γ, where γ is the Euler-Mascheroni constant (approx 0.57721...). And then that links back to the other infinite sum mentioned in the video. The Euler-Mascheroni constant is also the limit difference between the harmonic sum to X terms and ln(X). It's not too difficult to show that link algebraically.

@ekadria-bo4962 5 жыл бұрын

Achiled and toytoyss. Where is James Grime?

@ShantanuAryan67 5 жыл бұрын

ba na na oh na na ...

@Smokin438 4 жыл бұрын

This video is fantastic, more please

@NatetheAceOfficial 5 жыл бұрын

The animations for this episode were fantastic!

@InMyZen 5 жыл бұрын

loved this video, I coded the infinite series while going along with the video, cool stuff.

@rakhimondal5949 5 жыл бұрын

Those animations help to get the concept more clearly

@sasisarath8675 4 жыл бұрын

I love the way he handled the infinity question !

@lornemcleod1441 4 жыл бұрын

This is great, I'm learning about these I my Cal II class, and this just deepens my understanding of the infinite sums and series

@mrnarason 5 жыл бұрын

He's explanation is very much lucid. Being a fields medalist must be incredible.

@austynhughes134 5 жыл бұрын

Just another fantastic episode of Numerphile

@bachirblackers7299 3 жыл бұрын

Very smooth and lovely

@blogginbuggin 2 жыл бұрын

You've made Math fun. Thank you.

@rintintin3622 5 жыл бұрын

Surprising! Btw, I like your animations. Could you do a Numberphile-2 on how you make them?

@EddyWehbe 5 жыл бұрын

The last result blew my mind. I hope they show the proof in a future video.

@user-ct1ns6zw4z 5 жыл бұрын

Probably too hard of a proof for a numberphile video. 3blue1brown has a video on it though.

@hassanakhtar7874 4 жыл бұрын

@@user-ct1ns6zw4z nah I think you really can if you simplify Euler's first proof which was already a little hand-wavy.

@asdfghj7911 5 жыл бұрын

What a coincidence that you would post a video with Charles Fefferman today. I just handed in my dissertation which was on his disproof of the disc conjecture.

@jessecook9776 5 жыл бұрын

I just finished teaching about infinite series with my students in calculus 2. Sharing with my students!

@citrusblast4372 5 жыл бұрын

I remember this from pre cal :D

@justzack641 5 жыл бұрын

The fact they're using a different type of paper disturbs me

@mauz791 5 жыл бұрын

And it switches for the animations as well. Dammit.

@electrikshock2950 5 жыл бұрын

I like this professor , you can see that he loves what he's doing and is enthused about it but he doesn't let it get in the way of him explaining

@1959Edsel 5 жыл бұрын

This is the best explanation I've seen of why the harmonic series diverges.

@blitziam3585 5 жыл бұрын

Very interesting, thank you! You earned a subscriber.

@adammullan5904 5 жыл бұрын

I was convinced that Numberphile already had a video on all this, but I think I've just seen Matt Parker and VSauce both do it before...

@joeyknotts4366 5 жыл бұрын

I think numberphile has done it... I think it was not Matt Parker, but the red headed British mathematician.

@mathyoooo2 5 жыл бұрын

@@joeyknotts4366 James Grime?

@joeyknotts4366 5 жыл бұрын

@@mathyoooo2 ye

@samharper5881 5 жыл бұрын

And Vsauce doesn't know the difference between lay and lie so he doesn't matter anyway

@adammullan5904 5 жыл бұрын

Sam Harper that’s pretty prescriptivist of you tbh

@stormsurge1 5 жыл бұрын

I think you mixed up two Zeno's paradoxes, Achilles and the Tortoise and Dichotomy paradox.

@jerry3790 5 жыл бұрын

To be fair, he’s a fields medalist, not a person who studies Greek philosophers

@SirDerpingston 5 жыл бұрын

@@jerry3790 ...

@gregoryfenn1462 5 жыл бұрын

I was thinking that too.. does thus channel not have editors to do proof-read this stuff?????

@silkwesir1444 5 жыл бұрын

as far as I can tell they are very much related and it may be reasonable to bunch them together, as not two distinct paradoxes but two versions of the same paradox.

@muralibhat8776 5 жыл бұрын

@@gregoryfenn1462 this is a math channel mate. proof read what? achillies and the tortoise talks about the same problem as zeno's paradox of dichotomy

@oscarjeans4119 5 жыл бұрын

I like this guy! I hope he appears more often!

@HomeofLawboy 5 жыл бұрын

When I saw Infinite Series in the title my heart skipped a beat because I thought it was the channel infinite Series being revived.

@guangjianlee8839 5 жыл бұрын

We do need Pbs Infinite Series back

@ekadria-bo4962 5 жыл бұрын

Agree with you..

@michaelnovak9412 5 жыл бұрын

What happened to PBS Infinite Series is truly a tragedy. Honestly it was my favorite channel on KZfaq.

@tanishqbh 5 жыл бұрын

I thought infinite series was still kicking. What happened?

@michaelnovak9412 5 жыл бұрын

@@tanishqbh The hosts wanted to continue but PBS refused to continue supporting the channel, so it was closed down.

@skarrambo1 5 жыл бұрын

It's too late for an April Fools; where's the BROWN?!

@paulpantea9521 5 жыл бұрын

This guy is a genius. Please have more with him!

@eugene7518 26 күн бұрын

The genius forgot to mention that the tortoise is always moving forward like Achilles is.

@jriceblue 5 жыл бұрын

Your graphics person has the patience of a saint.

@Liphted 5 жыл бұрын

I didn't know Peter Shiff had a number channel!!! This is great!

@XRyXRy 5 жыл бұрын

Awesome, we're leaning about this in AP Calc!

@solandge36 4 жыл бұрын

This video creeped in when I was least expecting it.

@randomaccessfemale 5 жыл бұрын

What a cliffhanger! We are hoping that this pi occurrence will be explained in Infinite Series 2.

@ameyaparanjpe6179 5 жыл бұрын

great video

@uvsvdu 5 жыл бұрын

Charles Fefferman! I met him and his also very talented daughter last summer at an REU!

@winkey1303 Жыл бұрын

Thank you

@chessandmathguy 5 жыл бұрын

I just love that the p series with a p of 2 converges to pi^2/6.

@hcsomething 5 жыл бұрын

Is the Harmonic Series the series with the smallest individual terms which still diverges? Or is there some series of terns S_n where 0.5^n < S_n < H_n where the sum of S_n diverges?

@grovegreen123 5 жыл бұрын

really like this guy

@robinc.6791 5 жыл бұрын

Series was the hardest part of calc 2 :( but it makes sense now :)

@bobbysanchez6308 5 жыл бұрын

“And that’s one, thank you.”

@mikeandrews9933 5 жыл бұрын

My first encounter with the overhang question was from Martin Gardner’s “Mathematical Games” column of Scientific American. I used to do this all the time with large stacks of playing cards

@RobinSylveoff 5 жыл бұрын

6:43 “for a large enough value of a gazillion”

@nikitabelousov5643 4 жыл бұрын

animation is a blast!

@apolotion 5 жыл бұрын

Just took a calculus quiz that required us to use the comparison theorem to prove that the integral from 1 to infinity of (1-e^-x)/(x^2)dx is convergent. I happened to watch this just before taking the quiz and essentially saw it from a different approach. Numberphile making degrees over here 😂

@jerry3790 5 жыл бұрын

Wow! A fields medalist!

@fanemnamel6876 5 жыл бұрын

this ending... best cliffhanger ever!

@johnwarren1920 5 жыл бұрын

Nice presentation, but please don't use the wiggly (orange) numbers effect. It just makes it hard to read.

@rosiefay7283 5 жыл бұрын

I agree. Your constantly flickering text made the video unwatchable. -1. Please, Numberphile, never do this again.

@richardparadox7309 5 жыл бұрын

wiggly orange 🍊

@randomdude9135 5 жыл бұрын

Wiggly orange 🍊

@uwuifyingransomware 4 жыл бұрын

Wiggly orange 🍊

@denyraw 4 жыл бұрын

wiggly orange 🍊

@ShahryarKhan-KHANSOLO- 5 жыл бұрын

Great!

@doodelay 5 жыл бұрын

The series of comments in this thread converge on one conclusion and that is to Bring back PBS Infinite Series!

@eydeet914 5 жыл бұрын

Interesting new editing style and I believe theres lots of work behind it but I personally think I prefer the more static style. I was very distracted by all the wobbling (and the wrong kind of paper :D ).

@TaohRihze 5 жыл бұрын

So if 1/N^1 diverges, and 1/N^2 is bounded. So at which power between 1 and 2 does it switch from bounded to diverging?

@SlingerDomb 5 жыл бұрын

at exactly 1 well, you can study this topic named "p-series" if you want to.

@Anonimo345423Gamer 5 жыл бұрын

As soon as 1/n^a has an a>1 it converges

@josephsaxby618 5 жыл бұрын

1, if k is greater than 1, Σ1/n^k converges. If k is less than or equal to 1, Σ1/n^k diverges.

@SamForsterr 5 жыл бұрын

Taoh Rihze If k is any real number greater than one, then the sum of 1/N^k converges

@lagomoof 5 жыл бұрын

sum of n from 1 to infinity of 1/n^k converges for all k > 1. So there's no answer to your question because there's no 'next' real number greater than 1, but any number greater than 1 will do. k=1+1/G64 where G64 is Graham's Number will result in convergence, for example. But if you attempt to compute the limit iteratively it might take some time.

@emdash8944 5 жыл бұрын

Every math professor has their own word for a really big number.

@HackAcadmey 5 жыл бұрын

I like the Animation in this one

@deblaze666 5 жыл бұрын

For a large enough values of a gazillion

@jasonli1060 5 жыл бұрын

I am so shook

@sanauj15 5 жыл бұрын

interesting, I was just learning about series and sequences in my class today.

@micheljannin1765 5 жыл бұрын

This vid felt like Déjà-vu

@MrCrashDavi 5 жыл бұрын

VSAuce did it. We'll run out of edutainment before 2025, and there'll probably be mass suicides.

@mrnarason 5 жыл бұрын

Infinite series had been cover many times on this channel and others.

@trevorallen3212 5 жыл бұрын

Planck length is the minimal level before quantum physics starts extremely affecting the space time itself in those infintismal scales... Dam you zeno you did it again!!

@AdamDane 5 жыл бұрын

Pouring one out for PBS Infinite Series

@adnanchaudhary5905 3 жыл бұрын

The pile of cards with harmonic series is depicted in a math book. I've been searching for that book since a few years. I downloaded and read some part of it back in 2017. I lost it somehow and now I don't remember the name of the book. Is there anyone who knows the name of that book? Thanks!

@ianmoore5502 5 жыл бұрын

It took me 2 seconds to fall in love with his voice. Reminds me of M. A. Hamelin.

@kabirvaidya1791 5 жыл бұрын

This video should have been realeased 6 months ago when this was in my first year BSc 1sem portion

@SKhan-tb5zk 5 жыл бұрын

Where can I buy them books that appear in the video at time 4:15 s

@sander_bouwhuis 5 жыл бұрын

You stopped the video at the moment I thought it was getting interesting!

@jeffo9396 5 жыл бұрын

It was interesting from the very beginning.

@peterbreedveld1595 5 жыл бұрын

Do the books stacked on top of each other form a parabola?

@seikomyazawa 5 жыл бұрын

Where can I buy "the annals of mathematics"?

@lm58142 7 ай бұрын

The 1st infinite series mentioned corresponds to a different Zeno's paradox - that of dichotomy paradox.

@trelligan42 5 жыл бұрын

A phrase that illuminates the 'what does "sums to infinity" mean' is "grows without bound".

@koenth2359 5 жыл бұрын

In Zeno's version, the tortus is given a head start, but also walks, albeit slowlyer than Achilles. The point is that A runs to the starting point of T, but T is not there anymore, and next A has to run to where T is now, etc. So each step is smaller in a geometric series, but not necessarily one with ratio 1/2.

@priyankanarula5454 4 жыл бұрын

in which category you place infinity number theort

@manual1415 5 жыл бұрын

He looks so wholesome!

@navneetmishra3208 5 жыл бұрын

next video will be about pi square over 6?

@user-rd7jv4du1w 5 жыл бұрын

Naruto is an example of an infinite series

@noverdy 5 жыл бұрын

More like graham's number of series

@tails183 5 жыл бұрын

Pokémon and One Piece lurk nearby.

@lowlize 5 жыл бұрын

You mean Boruto's dad?

@NoNameAtAll2 5 жыл бұрын

Naruto ended Boruto began

@evanmurphy4850 5 жыл бұрын

@@noverdy Graham's number is smaller than infinity...

@vanhouten64 5 жыл бұрын

-1/12 is my favorite series

@Bobbymays 5 жыл бұрын

Its a lie

@Euquila 5 жыл бұрын

The fact that PI creeps in means that infinite series can be re-cast into some 2-dimensional representation (since circles are 2-dimensional). In fact, 3Blue1Brown did a video on this

@shiroshiro2183 5 жыл бұрын

Brilliance of S. Ramanujan infinite series

@charlesfort6602 5 жыл бұрын

So, if we add the surace area of a infinite series of squares, which sides lenght are the numbers of harmonic series, then we will get a finite surface area of pi^2/6, which also can be presented as a circle. (also the sum of their circuts will be infinite)

@57op 5 жыл бұрын

“..for a large enough value of gazillion..” 🤣

@XenoTravis 5 жыл бұрын

Vsauce and Adam Savage did a cool video a while ago where they made a big harmonic stack

@ashcoates3168 5 жыл бұрын

Travis Hunt KZfaq PhD what’s the video called? I’m interested in it

@VitaliyCD 5 жыл бұрын

@@ashcoates3168 Leaning Tower of Lire

@Jixzl 5 жыл бұрын

I remember the anals of mathematics. My lecturer gave it to me last semester.

@akosbakonyi5749 5 жыл бұрын

I guess he had a long ruler, heh?

@nocturnomedieval 5 жыл бұрын

Would like to see a series of vidss about series...so meta

@unneccry2222 5 жыл бұрын

numberfile infinite series colab!!!!

@navneetmishra3208 5 жыл бұрын

Centre of mass thing was awesome

@angharadhafod 5 жыл бұрын

Could the Tau adherents please explain this one?

@kevina5337 5 жыл бұрын

Nice video as always but kindof an abrupt ending. Some more details and discussion on the whole (pi^2)/6 thing would've been most welcome LOL

@lukaszrakus Ай бұрын

I wonder. How far on the x axies will "volume 1" move for the Grahams Number of terms?

@endermage77 5 жыл бұрын

2:14: TREE(3): *Allow me to introduce myself,*

@user-hz3sp8ns3p 5 жыл бұрын

In the end I was so hyped to see the proof that the last series equals pi^2/6, but not this day)

@lucbourhis3142 5 жыл бұрын

The lower bound used to show the harmonic series diverge is a pleasant trick but it does not tell us how fast the series diverge: the sum of the first n terms goes as the logarithm of n. We can even go further: it goes like log n plus the Euler constant plus a term behaving as 1/n. But that requires methods beyond mere arithmetic.