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.
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.
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
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.
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.
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.
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.
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.
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.
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.
@@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.
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.
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.
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.
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.
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 😂
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.
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).
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
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.
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
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.
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!!
infinity is possibility (in - finity) in something, between something - there are possibilities to definition (expression) space for existence - defined
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.
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 ).
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.
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.
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.
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)
Can someone explain leading upto 1:10 eventually the distance is 2....as far as I understand, it never get to 2 if u always halving the distance.... Right?
It gets to 2 after an infinite amount of steps. "Never" would imply that the series could not end in a finite amount of time, but if for each summand that you ad you take the proportional time unit, then an infinite series can end in a finite amount of time. If we take 1/2 a second to add the first summand, 1/4 of a second to add the second summand, 1/8 of a second to add the third summand etc. We still would have an infinite amount of summands to add/operations to perform, but it would converge in a finite amount of time. Hope that made sense.
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?
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Hello, you can't end it like that. Not without explaining how it becomes Pi^2/6
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.
Agreed! Let's hope they follow it up with another video.
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.
He also fails to make an argument for why he thinks that the first series ends up as equal to 2.
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
Introducing the Numberphile video channel which absolutely will never, ever be discontinued - The Infinite Series.
RIP the PBS RUclips channel of the same name :(
Exactly - that was one of my favorite channels.
What happened to PBS Infinite Series is truly a tragedy. It was my favorite channel on RUclips honestly.
PBS Infinite Series being discontinued wasn't much of a loss if you ask me.
@@-Kerstin Why? Did you find anything wrong with it?
can I just compliment the animations in this video? in terms of presentation, numberphiles has come such a long way, and I love it!
Props to the animator.
I miss the days of simple shorn parchment and sharpie.. 😔
My mind cannot handle the different kind of paper!
Yep. Stands out like a sore thumb.
It seems they had a shortage of brown paper rolls and decided to use brown envelopes instead!
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.
Its okay to be autistic
@@BloodSprite-tan well it's a lot flipping closer to brown than white!
Matt Parker used this series and equation to calculate pi on pi-day with multiple copies of his book named Humble Pi.
of course he did, haha
Ah yes I remember how he got 3.4115926...
@Perplexion Dangerman wait what
@@frederf3227 yes... 3.411....
@Perplexion Dangerman ~arrogance~
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.
Glad it's appreciated! Thanks
"PI creeps in where you would least expect it..." and so does this video.
false.
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.
I'm stealing this😎
lol poor mathematicians
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.
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.
Cool!
Achiled and toytoyss.
Where is James Grime?
ba na na oh na na ...
The fact they're using a different type of paper disturbs me
And it switches for the animations as well. Dammit.
The last result blew my mind. I hope they show the proof in a future video.
Probably too hard of a proof for a numberphile video. 3blue1brown has a video on it though.
I just finished teaching about infinite series with my students in calculus 2. Sharing with my students!
I remember this from pre cal :D
When I saw Infinite Series in the title my heart skipped a beat because I thought it was the channel infinite Series being revived.
We do need Pbs Infinite Series back
Agree with you..
What happened to PBS Infinite Series is truly a tragedy. Honestly it was my favorite channel on RUclips.
I thought infinite series was still kicking. What happened?
@@tanishqbh The hosts wanted to continue but PBS refused to continue supporting the channel, so it was closed down.
The animations for this episode were fantastic!
He's explanation is very much lucid. Being a fields medalist must be incredible.
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.
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.
@@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.
@@Xonatron Yea. Instead of triangles I've seen it with a square example, adding up to 2 square area units.
I think you mixed up two Zeno's paradoxes, Achilles and the Tortoise and Dichotomy paradox.
To be fair, he’s a fields medalist, not a person who studies Greek philosophers
@@jerry3790 ...
I was thinking that too.. does thus channel not have editors to do proof-read this stuff?????
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.
@@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
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...
I think numberphile has done it... I think it was not Matt Parker, but the red headed British mathematician.
@@joeyknotts4366 James Grime?
@@mathyoooo2 ye
And Vsauce doesn't know the difference between lay and lie so he doesn't matter anyway
Sam Harper that’s pretty prescriptivist of you tbh
This guy is a genius. Please have more with him!
The genius forgot to mention that the tortoise is always moving forward like Achilles is.
I love the way he handled the infinity question !
It's too late for an April Fools; where's the BROWN?!
Those animations help to get the concept more clearly
Nice presentation, but please don't use the wiggly (orange) numbers effect. It just makes it hard to read.
I agree. Your constantly flickering text made the video unwatchable. -1. Please, Numberphile, never do this again.
wiggly orange 🍊
Wiggly orange 🍊
Wiggly orange 🍊
wiggly orange 🍊
This is the best explanation I've seen of why the harmonic series diverges.
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
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.
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?
at exactly 1
well, you can study this topic named "p-series" if you want to.
As soon as 1/n^a has an a>1 it converges
1, if k is greater than 1, Σ1/n^k converges. If k is less than or equal to 1, Σ1/n^k diverges.
Taoh Rihze If k is any real number greater than one, then the sum of 1/N^k converges
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.
This video is fantastic, more please
Your graphics person has the patience of a saint.
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.
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 😂
loved this video, I coded the infinite series while going along with the video, cool stuff.
What a cliffhanger! We are hoping that this pi occurrence will be explained in Infinite Series 2.
6:43 “for a large enough value of a gazillion”
The series of comments in this thread converge on one conclusion and that is to Bring back PBS Infinite Series!
3:36 he really does mean that. You have to get a denominator of 272,400,599 just to get past 20 (20.000000001618233)
I remember the anals of mathematics. My lecturer gave it to me last semester.
I guess he had a long ruler, heh?
Naruto is an example of an infinite series
More like graham's number of series
Pokémon and One Piece lurk nearby.
You mean Boruto's dad?
Naruto ended
Boruto began
@@noverdy Graham's number is smaller than infinity...
I like this guy! I hope he appears more often!
Just another fantastic episode of Numerphile
You've made Math fun. Thank you.
Series was the hardest part of calc 2 :( but it makes sense now :)
I just love that the p series with a p of 2 converges to pi^2/6.
This vid felt like Déjà-vu
VSAuce did it.
We'll run out of edutainment before 2025, and there'll probably be mass suicides.
Infinite series had been cover many times on this channel and others.
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
“And that’s one, thank you.”
A phrase that illuminates the 'what does "sums to infinity" mean' is "grows without bound".
Vsauce and Adam Savage did a cool video a while ago where they made a big harmonic stack
Travis Hunt RUclips PhD what’s the video called? I’m interested in it
@@ashcoates3168 Leaning Tower of Lire
Charles Fefferman! I met him and his also very talented daughter last summer at an REU!
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.
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).
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
This video creeped in when I was least expecting it.
6:51 that suspiciously looks like half of a parabola... is it?
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)
thank you for this great video
Very interesting, thank you! You earned a subscriber.
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.
For a large enough values of a gazillion
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
The pi^2/6 comes from the Riemann Zeta function right?
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.
I was also thinking that
Euler soved this problem first
He forgot to mention that the tortoise is also always moving forward
Every math professor has their own word for a really big number.
Very smooth and lovely
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!!
Brilliance of S. Ramanujan infinite series
It took me 2 seconds to fall in love with his voice. Reminds me of M. A. Hamelin.
The square next to 1/20 is misplaced at 8:50 :P
nice, didnt see that
Well spotted! - I think that must have been the point where I realised how long it was going to take to finish...
Wow! A fields medalist!
You have had Villani, Tao, and now Fefferman. Would be amazing if you could manage to get Perelman on the show
are you joking?
The 1st infinite series mentioned corresponds to a different Zeno's paradox - that of dichotomy paradox.
infinity is possibility (in - finity) in something, between something - there are possibilities to definition
(expression) space for existence - defined
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.
2:14:
TREE(3): *Allow me to introduce myself,*
-1/12 is my favorite series
Its a lie
In the end I was so hyped to see the proof that the last series equals pi^2/6, but not this day)
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 ).
You stopped the video at the moment I thought it was getting interesting!
It was interesting from the very beginning.
Awesome, we're leaning about this in AP Calc!
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.
You would most likely see him on Sixty Symbols, Numberphile's physics-based sister channel.
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.
I didn't know Peter Shiff had a number channel!!! This is great!
The video ended right when it got interesting
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.
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)
The second series grows super slowly by the time you get to 1/1,000,000 you'll have only got to 14.39
Can someone explain leading upto 1:10 eventually the distance is 2....as far as I understand, it never get to 2 if u always halving the distance.... Right?
It gets to 2 after an infinite amount of steps. "Never" would imply that the series could not end in a finite amount of time, but if for each summand that you ad you take the proportional time unit, then an infinite series can end in a finite amount of time.
If we take 1/2 a second to add the first summand, 1/4 of a second to add the second summand, 1/8 of a second to add the third summand etc. We still would have an infinite amount of summands to add/operations to perform, but it would converge in a finite amount of time. Hope that made sense.
This guy is a natural born teatcher. He makes it very easy to understand.
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?
Pouring one out for PBS Infinite Series
Thank you
This video should have been realeased 6 months ago when this was in my first year BSc 1sem portion
Little did Achilles know, he just had to define his goal as what he was already doing!
really like this guy
Surprising! Btw, I like your animations. Could you do a Numberphile-2 on how you make them?
What about 1/3+1/9+1/27+1/81... and 1/9+1/81+1/729+1/6561...?
animation is a blast!
He forgot to credit Euler for his discovery of the solution to the third infinite series example discussed in this video.
Yes that pile will reach however far you want, but you also need to prove that the pile won’t collapse by doing so!
this ending... best cliffhanger ever!
For the next one you can show that the series 1/2 + 1/3 + 1/5 + 1/7 + ... +1/p + ... also diverges.
I like the Animation in this one