Infinite Series - Numberphile

[@snowgw2]

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

[@4dragons632]

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]

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

[@ipassedtheturingtest1396]

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]

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

[@andretimpa]

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

[@JJ-kl7eq]

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

[@b3z3jm3nny]

RIP the PBS KZbin channel of the same name :(

[@JJ-kl7eq]

Exactly - that was one of my favorite channels.

[@michaelnovak9412]

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

[@-Kerstin]

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

[@johanrichter2695]

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

[@Kilroyan]

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]

Props to the animator.

[@lazertroll702]

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

[@ruhrohraggy1313]

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]

I'm stealing this😎

[@bo-dg3bh]

lol poor mathematicians

[@ilyrm89]

My mind cannot handle the different kind of paper!

[@debayanbanerjee]

Yep. Stands out like a sore thumb.

[@rebmcr]

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

[@BloodSprite-tan]

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]

Its okay to be autistic

[@rebmcr]

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

[@erumaayuuki]

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

[@incription]

of course he did, haha

[@frederf3227]

Ah yes I remember how he got 3.4115926...

[@Danilego]

@Perplexion Dangerman wait what

[@InDstructR]

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

[@brennonstevens467]

@Perplexion Dangerman ~arrogance~

[@zuzusuperfly8363]

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]

Glad it's appreciated! Thanks

[@CCarrMcMahon]

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

[@Triantalex]

false.

[@maxpeeters8688]

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]

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.

[@UpstreamNL]

Cool!

[@justzack641]

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

[@mauz791]

And it switches for the animations as well. Dammit.

[@HomeofLawboy]

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]

We do need Pbs Infinite Series back

[@ekadria-bo4962]

Agree with you..

[@michaelnovak9412]

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

[@tanishqbh]

I thought infinite series was still kicking. What happened?

[@michaelnovak9412]

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

[@ekadria-bo4962]

Achiled and toytoyss. Where is James Grime?

[@ShantanuAryan67]

ba na na oh na na ...

[@jessecook9776]

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

[@citrusblast4372]

I remember this from pre cal :D

[@adammullan5904]

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]

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

[@mathyoooo2]

@@joeyknotts4366 James Grime?

[@joeyknotts4366]

@@mathyoooo2 ye

[@samharper5881]

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

[@adammullan5904]

Sam Harper that’s pretty prescriptivist of you tbh

[@NatetheAceOfficial]

The animations for this episode were fantastic!

[@EddyWehbe]

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

[@user-ct1ns6zw4z]

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

[@Xonatron]

0:51 the story according to the paradox is the tortoise is not caught *because* an infinite number of things have to happen and therefore never happen.

[@scepgineer]

The paradox was proven to be a falsidical paradox once we discovered calculus. Say one covers 1 length unit in 1 unit of time, then 1/2 length unit in 1/2 of a time unit, then 1/4 length unit in 1/4 of a time unit, then 1/8 length unit in 1/8 of a time unit and so on. Effectively traveling with 1 unit of velocity, covering a distance of 2 length units in 2 units of time. "would never happen" would imply that it is not possible to mathematically do what I described above. Granted this is only a mathematical problem not a problem of physics, since in the physical world there comes a point where spacetime can't be meaningfully divided beyond the Planck units.

[@Xonatron]

@@scepgineer Exactly. Calculus solved it. One way to visualize it is the area of a triangle, where you only move half way towards the 'tail' of it, counting up each area piece (don't bother with the math; just a visual exercise), never getting to the end, although the answer is finite and known.

[@scepgineer]

@@Xonatron Yea. Instead of triangles I've seen it with a square example, adding up to 2 square area units.

[@mrnarason]

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

[@skarrambo1]

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

[@stormsurge1]

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

[@jerry3790]

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

[@SirDerpingston]

@@jerry3790 ...

[@gregoryfenn1462]

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

[@silkwesir1444]

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]

@@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

[@johnwarren1920]

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

[@rosiefay7283]

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

[@richardparadox7309]

wiggly orange 🍊

[@randomdude9135]

Wiggly orange 🍊

[@uwuifyingransomware]

Wiggly orange 🍊

[@denyraw]

wiggly orange 🍊

[@sasisarath8675]

I love the way he handled the infinity question !

[@paulpantea9521]

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

[@eugene7518]

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

[@electrikshock2950]

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

[@TaohRihze]

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]

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

[@Anonimo345423Gamer]

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

[@josephsaxby618]

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]

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

[@lagomoof]

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.

[@asdfghj7911]

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.

[@rakhimondal5949]

Those animations help to get the concept more clearly

[@1959Edsel]

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

[@user-rd7jv4du1w]

Naruto is an example of an infinite series

[@noverdy]

More like graham's number of series

[@tails183]

Pokémon and One Piece lurk nearby.

[@lowlize]

You mean Boruto's dad?

[@NoNameAtAll2]

Naruto ended Boruto began

[@evanmurphy4850]

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

[@koenth2359]

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.

[@apolotion]

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 😂

[@Smokin438]

This video is fantastic, more please

[@randomaccessfemale]

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

[@jriceblue]

Your graphics person has the patience of a saint.

[@doodelay]

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

[@InMyZen]

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

[@zperk13]

3:36 he really does mean that. You have to get a denominator of 272,400,599 just to get past 20 (20.000000001618233)

[@Jixzl]

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

[@akosbakonyi5749]

I guess he had a long ruler, heh?

[@zperk13]

2:00 i wrote some code to see how long it would take to get to a number. I am not going to do 50 trillion as that would take a way too long, so I will do 20. You might be thinking I'm making it to easy but I tried other numbers and they were just taking too long. To get to 20 you would need the denominator to get to 272,400,601. That took 27 seconds to computer. For comparison, it took half a second to calculate 16, and it took 76 seconds to calculate 21. 740,461,602 is the denominator you have to get to to reach 21 btw.

[@micheljannin1765]

This vid felt like Déjà-vu

[@MrCrashDavi]

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

[@mrnarason]

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

[@Euquila]

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

[@hcsomething]

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?

[@lornemcleod1441]

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

[@WindowsXP_YT]

What about 1/3+1/9+1/27+1/81... and 1/9+1/81+1/729+1/6561...?

[@trelligan42]

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

[@uvsvdu]

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

[@oscarjeans4119]

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

[@XenoTravis]

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

[@ashcoates3168]

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

[@VitaliyCD]

@@ashcoates3168 Leaning Tower of Lire

[@austynhughes134]

Just another fantastic episode of Numerphile

[@robinc.6791]

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

[@chessandmathguy]

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

[@RobinSylveoff]

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

[@deblaze666]

For a large enough values of a gazillion

[@laszlosimo788]

infinity is possibility (in - finity) in something, between something - there are possibilities to definition (expression) space for existence - defined

[@bobbysanchez6308]

“And that’s one, thank you.”

[@fearitselfpinball8912]

1 + 1/2 + 1/4 + 1/8… Every possible number in this series has the same two properties in common: A. It _diminishes_ the ‘gap’ (between the accumulating number and 2). B. It fails to close the gap between the accumulating number “2”. Since every possible number in the whole series is _incapable_ of closing the gap it diminishes, adding _all of the numbers_ (the ‘infinite sum’) does not involve adding any number which reaches 2. Achilles does not catch the Tortoise. Also, since the gap size (the distance between the accumulated number and 2) is the last number in the series (gap of 1/4 at 1+1/2+1/4) the accumulation of numbers can _never_ result in the closure of the gap.

[@Ralesk]

6:12 the lean isn't 1, 1/2, 1/3, ... but 1/2, 1/3, 1/4 and so on - doesn't change the end result (infinity minus one is still infinity), but the visualisation really bothered me there.

[@ruroruro]

Listen carefully to what he says. "The distances are in **proportions** 1, 1/2, ..." The listed numbers are proportions relative to the first overhang, not relative to the book length. The reason, he says it that way is because if you take the book length to be 1, the lengths of overhangs would be 1/2, 1/4, 1/6, 1/8, 1/10, 1/12, ... (half of the harmonic series).

[@mikeandrews9933]

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

[@blogginbuggin]

You've made Math fun. Thank you.

[@lm58142]

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

[@mariovelez578]

now someone please tell me why 1/nln(n) diverges. we can show it diverges through an integral test. as a p-series, 1/n barely diverges, whereas 1/(n^1.000...1) converges. why does multiplying the bottom by ln(n), a function where the lim as n -> ∞ = ∞, still not make it converge.

[@eydeet914]

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 ).

[@emdash8944]

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

[@trevorallen3212]

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!!

[@thomasjakobsen2260]

The pi^2/6 comes from the Riemann Zeta function right?

[@Arycke]

Yes and no. "Comes from" is vague. The problem where that result first appeared was called the Basel Problem posed about 80 years before Euler established a proof and about 200 years before Riemann published related results. It was then associated merely by Riemann's construction of his zeta function as it is of the form sum(1,inf, 1/n^s, Re(s)>1). Euler did more work generalizing the result 50 to 100 years before Riemann published his most iconic paper using Euler's work.

[@randomdude9135]

I was also thinking that

[@eugene7518]

Euler soved this problem first

[@eugene7518]

He forgot to mention that the tortoise is also always moving forward

[@blitziam3585]

Very interesting, thank you! You earned a subscriber.

[@jonathanguzman8584]

thank you for this great video

[@solandge36]

This video creeped in when I was least expecting it.

[@mauricereichert2804]

The square next to 1/20 is misplaced at 8:50 :P

[@kevinhart4real]

nice, didnt see that

[@pmcpartlan]

Well spotted! - I think that must have been the point where I realised how long it was going to take to finish...

[@Sicira]

6:51 that suspiciously looks like half of a parabola... is it?

[@comma_thingy]

I might be a bit late, but no. It's the shape of the log curve (inverse of exponential) rather than the sqrt curve (inverse of a parabola)

[@bachirblackers7299]

Very smooth and lovely

[@charlesfort6602]

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)

[@lucbourhis3142]

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.

[@divergentmaths]

If you are interested to learn more about divergent series and want to understand why and how 1+2+3+4+5+6+... = -1/12, I recommend the online course “Introduction to Divergent Series of Integers” on the Thinkific online learning platform.

[@rintintin3622]

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

[@ianmoore5502]

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

[@nekogod]

The second series grows super slowly by the time you get to 1/1,000,000 you'll have only got to 14.39

[@XRyXRy]

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

[@willb9159]

Could you possibly ask Ed Witten to talk on the channel; especially since he's a physicist with a Fields medal! He also lectures at Princeton, just like Prof. Fefferman.

[@Arycke]

You would most likely see him on Sixty Symbols, Numberphile's physics-based sister channel.

[@Liphted]

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

[@bradensorensen966]

Summing squares of 1/x where x increments each squared value is well-known to have a relationship with pi, though.

[@WonderingBros]

Dear Numberphile could you make a series for beginners in mathematics or a video on how to be mathematician without college degree and tell us about references helping us achieving such a big Goal

[@КимБадук]

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

[@ffggddss]

Some interesting variants of the harmonic series - The alternating harmonic series: 1 - ½ + ⅓ - ¼ + ⅕ - ⅙ + - ... = ln(2) The alternating odd harmonic series: 1 - ⅓ + ⅕ - ¹/₇ + - ... = ¼π Also, the 'tamed' harmonic series: 1 + ½ + ⅓ + ¼ + ⅕ + ⅙ + ... + 1/n - ln(n) → γ = 0.5772156649... [as n→∞] But these deserve another video... Fred

[@divergentmaths]

If you are interested to learn more about divergent series and want to understand why and how 1+2+3+4+5+6+... = -1/12, I recommend the online course “Introduction to Divergent Series of Integers” on the Thinkific online learning platform.

[@ScLuigi]

That's not Achilles and the tortoise. That's the dichotomy. Achilles and the tortoise has the tortoise moving at a rate slower than Achilles, whereas the dichotomy has a static finish line, which is what was described in this video.

[@disgruntledtoons]

For the next one you can show that the series 1/2 + 1/3 + 1/5 + 1/7 + ... +1/p + ... also diverges.

[@Frahamen]

sounds better than ((τ/2)²)/6....

[@krzysztof-michalak]

well that would be τ²/24 so not that bad

[@yaeldillies]

τ^2/24, you mean? :P

[@jimi02468]

But 6 is better than 24

[@krzysztof-michalak]

@@jimi02468 but Tau is better then Pi so it's a draw :)

[@silkwesir1444]

@@jimi02468 nah. 24 is the real deas. 6 is just a quarter of it.

[@jazzabighits4473]

But 1 + 1/2 + 1/4 + 1/8 ... would be

[@connorneely3458]

Yep that last part is exactly it

[@jazzabighits4473]

@@connorneely3458 Well that makes more sense then, thank you

[@cwaddle]

You have had Villani, Tao, and now Fefferman. Would be amazing if you could manage to get Perelman on the show

[@holliswilliams8426]

are you joking?

[@VictorZWL]

Yes that pile will reach however far you want, but you also need to prove that the pile won’t collapse by doing so!

[@derfunkhaus]

IMO it would be a more beautiful equation, instead of saying the series converges to pi squared over 6, to rearrange it to the square root of [(3!)(1+ 1/4 + 1/9 + 1/16 + 1/25...)] = pi. Then you've got a formula that, like the famous e^i*pi formula, incorporates several notable mathematical concepts including addition, fractions, square roots, and factorials. 3! is arguably the first non-trivial factorial, which makes that interesting. Or, you could just call it 6 and be satisfied that it is the least composite number with two distinct prime factors.

[@shiroshiro2183]

Brilliance of S. Ramanujan infinite series

[@davidwilkie9551]

It's a convenient way to illustrate the transverse Phys-Chem connection of modulated QM-Time Principle in the sum-of-all-histories form-ulae, all the incident points of view of the tangential Superspin big picture.., to show the temporal superposition logic of quantization...

[@vanhouten64]

-1/12 is my favorite series

[@Bobbymays]

Its a lie

[@phyarth8082]

1/x converges in physics tasks (only technically converges), when you get unit hyperbola y=1/x, and in this case you get number pie and e constant, but technically converges because 1/x algebraic parameters is from (0, infinity) yeah you can not divide number by zero, yeah in physics always get reminder not equal to zero, but physics are not same as mathematics.

[@kabirvaidya1791]

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

[@TemplerOO7]

Interesting how slowly the harmonic series diverges. Is there a way to approximate the value of the sum for a given number of steps?

[@angelmendez-rivera351]

Yugo Betrugo The sequence of harmonic numbers increases logarithmically. It diverges because the limit of ln(n) as n increases indefinitely is infinite, meaning there is no limit.

[@angelmendez-rivera351]

Also, there is no closed form formula without using special functions, but using special functions defeats the obvious purpose of closed forms.

[@MelindaGreen]

What's the function on the exact boundary between converging and diverging?

[@angelmendez-rivera351]

Melinda Green There is no such function. There cannot be such function due to the very definition of divergence.

[@MelindaGreen]

@@angelmendez-rivera351 What's the proof of that?

[@endermage77]

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

[@adnanchaudhary5905]

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!