700 years of secrets of the Sum of Sums (paradoxical harmonic series)

Today's video is about the harmonic series 1+1/2+1/3+... . Apart from all the usual bits (done right and animated :) I've included a lot of the amazing properties of this prototypical infinite series that hardly anybody knows about. Enjoy, and if you are teaching this stuff, I hope you'll find something interesting to add to your repertoire!
00:00 Intro
01:00 Chapter 1: Balanced warm-up
03:26 Chapter 2: The leaning tower of maths
12:03 Chapter 3: Finite or infinite
15:33 Chapter 4: Terrible aim
20:44 Chapter 5: It gets better and better
29:43 Chapter 6: Thinner and thinner
42:54 Kempner's proof animation
44:22 Credits
Here are some references to get you started if you'd like to dig deeper into any of the stuff that I covered in this video. Most of these articles you can read for free on JSTOR.
Chapter 2: Leaning tower of lire and crazy maximal overhang stacks
Leaning Tower of Lire. Paul B. Johnson American Journal of Physics 23 (1955), 240
Maximum overhang. Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, Uri Zwick arxiv.org/abs/0707.0093
Worm on a rubber band paradox: en.wikipedia.org/wiki/Ant_on_...
Chapter 3: Proof of divergence
Here is a nice collection of different proofs for the divergence of the harmonic series scipp.ucsc.edu/~haber/archives...
Chapter 4: No integer partial sums
A harmonikus sorrol, J. KUERSCHAK, Matematikai es fizikai lapok 27 (1918), 299-300
Partial sums of series that cannot be an integer. Thomas J. Osler,
The Mathematical Gazette 96 (2012), 515-519
Representing positive rational numbers as finite sums of reciprocals of distinct positive integers www.math.ucsd.edu/~ronspubs/64...
Chapter 5: Log formula for the partial sums and gamma
Partial Sums of the Harmonic Series. R. P. Boas, Jr. and J. W. Wrench, Jr.
The American Mathematical Monthly 78 (1971), 864-870
Chapter 6: Kempner's no 9s series:
Kempner in an online comic
www.smbc-comics.com/comic/mat...
A very nice list of different sums contained in the harmonic series en.wikipedia.org/wiki/List_of...
Sums of Reciprocals of Integers Missing a Given Digit, Robert Baillie, The American Mathematical Monthly 86 (1979), 372-374
A Curious Convergent Series. A. J. Kempner, The American Mathematical Monthly 21 (1914), 48-50
Summing the curious series of Kempner and Irwin. Robert Baillie, arxiv.org/abs/0806.4410
If you still know how to read :) I recommend you read the very good book Gamma by Julian Havil.
Bug alert: Here • 700 years of secrets o... I say "at lest ten 9s series". That should be "at most ten 9s series"
Today's music (as usual from the free KZbin music library): Morning mandolin (Chris Haugen), Fresh fallen snow (Chris Haugen), Night snow (Asher Fulero), Believer (Silent Partner)
Today's t-shirt: rocketfactorytshirts.com/are-...
Enjoy!
Burkard
Two ways to support Mathologer
Mathologer Patreon: / mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)

[@daemos_magen]

The most memorable part was when you giggle, and my wife in the other room says "You're watching that math guy again?" As always, thank you for expanding my knowledge base.

[@piratesofphysics4100]

😂😂😂

[@douglasstrother6584]

When she hears all of the profanity, she knows you're watching Flammable Maths!

[@christiaanbalke]

Same over here :)

[@teleny2]

His giggling always sounds like Dr. Strangelove to me. Man, Peter Sellers was a great actor.

[@douglasstrother6584]

@@teleny2 Gen. Turgidson: "'Strangelove'? That ain't no kraut name." Aide: "His original name was 'Merkwürdigliebe'. He changed when he became a citizen." Gen. Turgidson: "Huh. Strange."

[@Fun_maths]

"Are we there yet?" "No just 1+1/2+1/3+1/4+... more minutes."

[@Mathologer]

:)

[@johnchessant3012]

In a similar vein, a mathematician advertises a lottery in which the prize is an infinite amount of money. Lots of people pay for tickets, but when the winning ticket is announced, the mathematician explains the mode of payment: "$1 this week, $1/2 next week, $1/3 the week after, ..."

[@MGSchmahl]

@@johnchessant3012 Given the common assumptions about compound interest and the time value of money, this prize can actually be funded with a finite amount of money. (The assumption is that $1 today is worth $e^(rt) at time t for some constant r.) A neat puzzle is to figure out how many "now dollars" that prize is worth. (Hint 1: If the prize was $1 every week, and assuming a realistically small interest rate like 1/52% per week, you would only need about $5200.50 to fund the prize.) (Hint 2: I don't know how to solve this by hand. I cheated and used WolframAlpha.) (Hint 3: It's surprisingly small! Less than $10.)

[@leif1075]

@@Mathologer Why isn't this more widely known? And how can we learn this faster?

[@Fun_maths]

i just realized that that means that they are pretty close to their destenation, about half a minute away

[@bernyelpro1906]

Most memorable part: me losing my life after failing the “no nines sum converges”

[@christianorlandosilvaforer3451]

sure.. x2

[@Torthrodhel]

I didn't lose my life at that part! I gamed the system, by already losing it way earlier on in the video! lmao

[@ireneonajarila4138]

@@Torthrodhel t

[@sergiomv85]

I lost my life too!

[@apbmes7690]

Most memorable part: In university Mathologer apparently came up with an original finiteness proof for Kempner's series, and the grader failed the homework because they couldn't be bothered to check a solution that was different from the one on the answer sheet.

[@IoEstasCedonta]

We've all had those graders.

[@snowstarsparkle]

[i

[@stopthephilosophicalzombie9017]

I had a similar experience in topology in undergrad. I did an unconventional proof and even my professor didn't understand it but he found another professor who said it was correct.

[@jetzeschaafsma1211]

How often do you imagine an answer different from the answer sheet is actually correct?

[@Meta7]

@@jetzeschaafsma1211 In math, more often than you think.

[@shivambiswas5790]

Any divergent series: *exists* Ramanujan: Allow me to make it convergent.

[@456MrPeople]

Ramanujan summation is powerful but it is also very picky. For example if I add 1+2+3+4+... and 0+1+2+3+4+... under Ramanujan summation they would have completely different values! Even if I add the numbers in a different order they would have different values. A small price to pay given the ability to sum divergent series.

[@przemysawkwiatkowski2674]

@@456MrPeople It's not so strange that order of summation changes the sum... It might happen even for convergent series. :-) Well... Actually it *does* happen for convergent series, except the absolutely convergent ones. :-)

[@Noname-67]

@@456MrPeople that's the normal problem of infinity

[@sharpfang]

...to something that is not even close to where the series goes.

[@EdoTimmermans]

It would like to see how TREE(1)+TREE(2)+TREE(3)+... can be made convergent. 😉

[@jzieba0204]

The most suprising part for me was the "terrible aim" the fact that odd/even is never an integer is so simple yet i would have never thought about it

[@PC_Simo]

Yes. Simple, indeed: In order for a fraction to be an integer, the prime factors of the denominator must form a subset of the prime factors of the numerator; but, in odd/even, the denominator always has the prime factor: 2, which the numerator never does, in odd/even; thus, P(denominator)* is never a subset of P(numerator); and thus, odd/even can never be an integer, in disguise 😌. *P = Prime factors.

[@n0nam3given]

As an adult who barely survived "New Math" back in the 60s, I grew to *hate* math with a purple passion, though I loved it with an equal passion. I gave up, finally, in high school at algebra 1, with the only "C" I've ever received in all my school years. I guess they were trying to tell me that math is not my shtick. Today, that hatred has melted away and my love and curiosity shine again. I never miss any of your videos. I love your humor and your enthusiasm!! The most memorable part? The searching for and recognition of patterns. That is so delicious!

[@Mathologer]

That's great, your comment made my day :)

[@sergeboisse]

Mathologer video series are definitely better than any Netflix series. They surprise me anytime.

[@Achrononmaster]

With a small amount of effort one could probably get Mathologer onto Netflix. It's just filling in forms and checking video quality and whatnot.

[@BritishBeachcomber]

Netflix? No comparison. Mathologer wins every time, and it's free.

[@manelmanolo7195]

Mathflix. The best series (Taylor, MacLaurin, armonic, ...) (Seen in his t-shirt)

[@greatawakeningforall]

Exactly true 👍

[@zacharystark5520]

Most memorable: that the harmonic series narrowly misses all integers by ever shrinking margins

[@TheM0JEC]

I agree that an infinite number of non intergers is quite amazing.

[@MasterHigure]

I mean, any diverging series with ever smaller terms will have ever shrinking margins (as long as it doesn't actually hit any integers).

[@landsgevaer]

@@MasterHigure I don't think so. For example, consider the sequence x_0 = 9/4 and for all n > 0: x_n = 1+(1/3)^n; form a series by summing these terms. The terms are ever decreasing, the series is divergent, and never hits any integers. Yet the partial sums never come closer than 1/4 to any integer, which it hits at the very first element a_0 alone.

[@MasterHigure]

@@landsgevaer You're right. The terms need to converge to 0. I done goofed.

[@parkershaw8529]

It also managed to miss infinitely more and infinitely denser all irrational numbers as well. THAT seems even more impressive!

[@thek3nger]

I liked A LOT that the sum of the “exactly 100 zeros series” is greater than the “no 9s series”! It is almost unbelievable. I need to check the paper. 🤣

[@imacds]

It is intuitively plausible as the 100 zeros series includes "a lot" of terms with 9s that the no 9s series leaves out.

[@jackw7714]

@@imacds I can't say that's intuitive to me 😂

[@stephenmcateer]

First thought: Isn't the "exactly 100 zeros" series a subseries of the "no 9s" series? Second thought: No, what about the term 1/9e100? Third thought: So ... when you are dealing with *all* integers you thin out more by banning 9s than by *only* requiring 100 zeros. Mind blown. Fourth thought: Take a random billion-digit number. It will almost always have more than 100 zeros (you expect about 10 million of them, just 99 is very rare). So it's almost not a constraint at all. Although apparently enough to force convergence. On the other hand, almost no billion-digit numbers will have no nines (the probably is something like (9/10)^1e9 ~= 0 of randomly grabbing one). And almost all integers are bigger than just a billion digits. Mind now thoroughly blown.

[@NoNameAtAll2]

how was the paper?

[@gregburnell8454]

This blew my mind!

[@whatby101]

Undergraduate mathematician here. The better I get at math, the more I appreciate your videos. These videos give a great visual experience which is generally not taught in proof courses. My favorite chapter was probably Chapter 5, reminded me of some of the concepts discussed in my analysis course.

[@dennisbell9639]

My wife viewed this lecture,I made her,and just called you the biggest nerd on the planet.But that is good for she has been calling me the biggest one for 37 years I gladly pass the title over to you.I thoroughly enjoyed it and love your enthusiasm.I'm self-studying figurate numbers and would enjoy any lectures on this subject matter.Thank you

[@dEntz88]

The most memorable proof is the original proof of the harmonic series' divergence simply for the fact that this probably the only proof I could present to my year 10 math class and most of them would understand it.

[@Mathologer]

Would be interesting what your kids would make of the animation of this proof :)

[@dEntz88]

@@Mathologer Maybe I'll use it in my "Mathe AG". :)

[@morodochable]

The fact that the number of fractions summing to one in that doubled every time hinted at the logarithmic relationship, although I was thinking log base 2.

[@Meepmows]

The most memorable part for me might be the idea of that gamma value: especially the super quick visual proof that it had to be less than one by sliding over all the blue regions to the left

[@chessnotchekrs]

Yeah that was really mind-blowing. Also, to answer Mathologer's question, Gamma is more than half because each time we slide over the blue part, there is a corresponding white part, but the blue part has a "belly", or it bulges into the white part, so they're not equally divided triangles. There are infinitely many blue-half/white-half pairs, and in each the blue part has a "belly" so adding the area of all the blue "halfs" should yield a sum slightly more than half. This is just a visual approximation though, I don't know how to prove how much more than half it is.

[@atimholt]

@@chessnotchekrs Yeah, that was a fun one to just suddenly get (though, like he said, it was “obvious”).

[@mattbox87]

Most memorable: being invited to take a moment and post why it might be obvious that gamma is greater than 0.5 and then doing it. Hmm... why is it obvious that gamma is greater than 0.5? Well it didn't seem obvious... But imagine the blue bits were triangular; then there would be equal parts blue and white in the unit square on the left i.e. a gamma of 0.5. But the blue parts are convex, they each take up more than half of their rectangles and together take up more than half of the square.

[@ffggddss]

Exactly how I pictured it! It also makes it obvious that γ is much closer to ½ than it is to 1. Fred

[@bagochips1208]

Most memorable part: all I’m just constantly being mind blown throughout the whole video

[@sadkritx6200]

Same bro. The whole video was magnificent 👌

[@justsomeguy5628]

The weirdest thing you showed is definitely the unusual optimal brick stacking pattern.

[@Igneous01]

I wonder if we could train a machine learning model to see if there exist further optimizations to this question. This solution looks similar to something a model would come up with.

[@maze7474]

Why is that optimal stack optimal? Those 3 bricks on the top right look like you could extend them more to the left and thereby push the whole center of gravity to the left and thereby the tower to the right

[@ramenandvitamins]

​@@maze7474 ​moving a few bricks would necessarily shift the entire stack's center of gravity by a smaller distance. Since the blocks you're proposing to shift include the rightmost one, you'd lose more overhang than you'd gain.

[@maze7474]

@@ramenandvitamins sorry, typo from my side.I meant top left, those 3 that are stacked exactly over each other

[@ramenandvitamins]

@@maze7474 I suspect they'd no longer suffice to hold down the second-rightmost block if they were moved any further left.

[@neomew]

Math: '*exists* Euler: "First!"

[@ashtonsmith1730]

so how did math work before euler?

[@neomew]

@@ashtonsmith1730 The same way it always did - just in a dark room. Euler just turned on the lights to a lot of rooms.

[@raphaelreichmannrolim25]

When you're investigating maths, if you inspect close, there are really small notes all over the place. "Euler has been here"!

[@andrewxc1335]

@@ashtonsmith1730 Lots of wordy descriptions and cobbled-together notations from dozens of different people.

[@davidrheault7896]

@@neomew you'll never kill the real Dumbledore hehe

[@danield1303]

For me, the "no integers" part was the most memorable, but honestly the whole video was of great quality (as expected).

[@I_like_math]

Most memorable part: the 100 zeros sum being larger than the no nines sum.

[@davidgustavsson4000]

I liked your evil mathematician back story, with the teacher refusing to grade the "wrong" proof.

[@amadeus1940]

I didn't like it. Second hand annoyance. grrr

[@ummerfarooq5383]

When you get to that age when you want your students to tell you a bedtime story of the old days via math proofs. It would be gracious of us to do so just like when we were little kids asking mommy for a bed time story. Hmm are tests care work? 👋🕊️

[@moroccangeographer8993]

I relate to that experience.

[@rexwhitehead8346]

Peter Ustinov relates that his teacher gave him zero marks when he answered "Rimsky-Korsakov" to the question "Name one Russian composer." The correct answer was Tchaikovsky.

[@colinwebster2455]

Most memorable moment was the cat going "μ".

[@teleny2]

Has a cat the hacker-nature? "Mew...."

[@dragifire]

Your teaching style is just so good! I think it's a combination of the interesting topics, your smooth as heck animations, giggles, and the quick glances you give at the end of each chapter to summarize (it's especially nice for note-taking!). Not even to mention the fact that you don't give direct answers to questions you bring up, but instead direct the viewer to introductory terms and topics to look up and gain knowledge themselves. I wish I could attend one of your lectures, but until then this will have to do!

[@trueriver1950]

The rapidity of the harmonic series converging to infinity makes the tortoise in the fable seem rather rapid

[@anon6514]

They should have called it the 'Barely Divergent Series'

[@takyc7883]

is t the slowest diverging series?

[@yazeed0ps3]

@@takyc7883 I remember that there is no such thing as a slowest diverging series. for example 1/n diverges, 1/(n*ln(n)) diverges, 1/(n*ln(n)*ln(ln(n))) diverges, and so on. As always, there's a math stack exchange thread talking about this topic: math.stackexchange.com/questions/452053/is-there-a-slowest-rate-of-divergence-of-a-series

[@tracyh5751]

@@takyc7883 No. Using the ideas from chapter 6, you can actually show there is a subseries of the harmonic series that diverges as slowly as you would like. Simply take a function f(x) that diverges at a rate slower than the natural logarithm. At each integer, we will choose an entry from the harmonic series which is smaller than the one we had chosen previously. First, choose the largest entry of the harmonic series that is smaller than f(1). If this is not possible, choose the smallest entry of the harmonic series with is larger. Next, choose the largest entry possible so that the partial sum so far (just the first term and this one) is less than f(2). Again, if this is not possible, choose the smallest entry possible so that the partial sum is larger than f(2). Continue in this way and you will make a series whose rate of divergence is the same as f(x).

[@karateoscar]

Most memorable: The harmonic series misses all integers up to infinity

[@Jack_Callcott_AU]

That got me too.

[@thatssomethingthathappened9823]

Yeah it must be ∞.438882647883976917983791870000364553678223... or something.

[@aegirson2899]

Most memorable: The most efficient overhanging structure being the weird configuration instead of an apparently more ordered one.

[@naimulhaq9626]

Most memorable: An overhanging structure with n=google bricks.

[@anselmobd]

Most memorable: “No nines sum... converges!” 8^O´´´

[@_kapy_]

The most memorable thing is how ugly the optimal leaning tower is

[@Sn0wjunk1e]

you mean beautiful

[@EdoTimmermans]

Some people say warthogs are ugly. To me they are stunningly beautiful, with their faces resembling the Mandelbrot fractal.

[@tim40gabby25]

If extended hugely, I guess that patterns of absent blocks will create pleasing curves.. I recall doing this with kids bored of 'Jehinga'. old uk duffer here :)

[@naimulhaq9626]

It was the most beautiful leaning tower.

[@ammaleslie509]

i thought it was beautiful also. Glad I'm not alone. Modern concepts of "beauty" are overfocused on symmetry. Observe more natural structures to appreciate the beauty in the "misshapen" and the perfection in the "imperfect"

[@jean-francoistremblay7744]

Clearly, the highlight of the Euler-Mascheroni constant is a splendid part of the video...the sum of no 9's animation is very impressive.

[@rohitjoshi2953]

Most Memorable : Every seconds of this video. I couldn't choose a single thing. I am sure that this is the best video I have ever watched in my life related to anything. Thank you so much Mathologer.

[@HyperFocusMarshmallow]

You’re so good at starting simple and yet including stuff that’s interesting for the fully initiated! Great work!

[@johnchessant3012]

24:30 It's "obvious" because 1/x is concave, meaning between any two points the graph is below the secant line connecting those two points. Dividing the 1x1 square into rectangles in the obvious way, the blue areas include more than half of each rectangle and hence more than half of the 1x1 square.

[@Mathologer]

Exactly :)

[@MonsieurBiga]

Finally something I had seen myself with my very low level of maths

[@anthonycousins853]

That makes sense! Good explanation, I got it without any visuals! Haha.

[@jisyang8781]

You mean convex. You triggered one of my pet peeves.

[@moritzalshuth7239]

The secant lines partition blue triangles as a lower bound for gamma, triangles add up as a telescoping sum 1/2*((1/1-1/2)+(1/2-1/3)+...-1/n) = 1/2*(1-1/n) = 1/2 in the limit

[@peter_p_r_zhang]

Most memorable part: derivation of γ. As in high school we learn about the approximation of the area under the 1/x curve but not many actually focus on the 'negligible part of the area' which in fact adds up to something trivial to the whole field of number series. 24:23 Sinple proof for γ>0.5: All the tiny little bits of that blue areas are a curved shape. By connecting the two ends of that curve line we can see each part is made up of a triangle and a curved shape. The total area of those infinitely many triangles equals to 0.5 so the total area of the blue sharpest be greater than 0.5.

[@Mathologer]

That's it:)

[@youssefm1]

Why is the total area of those triangles 0.5?

[@Ohhelmno]

@@youssefm1 it’s basically because the largest non-integer in the series is 1/2 and every subsequent one is half again, so the first few get you very close .5 and every one after that is less and less and therefor as the sum gets closer to infinity the area above the curve gets closer to .5 but never over. This is mainly because there are an even more infinite set of fractions between 1/2 and 1/∞ than integers between 1 and ∞

[@youssefm1]

@@Ohhelmno , thanks. My son made me realise that the sum of the vertical lines (heights) of all the triangles = 1 so the areas of the triangles (being half the area of the rectangle of that height) = 0.5 and since the blue part was larger than the triangle, its area > 0.5.

[@williamrutiser1485]

Don

[@danicorrales1592]

Most memorable moment: the posture problems due to excessive obsession with mathematics

[@eruwa1550]

The fact that the harmonic series misses all the integers is beautiful to me!

[@alexgreer878]

Most memorable: If my life depended on knowing if the sum of no nines series is finite I would not be alive

[@alejandronasifsalum8201]

I was already... not comfortable, but let's say "resigned"... to the fact that there exist very slowly divergent series; but the fact that there are very slowly CONVERGENT series, whose sum is impossible to approximate computationally within a reasonable margin of error, like the no nines series... was a shock!

[@elasiduo108]

This is the reason why I'm getting a PHD in mathematics: the infinite beauty of the numbers.

[@takeguess]

I have to say you are one of my favorite KZbinrs! And that is saying something.... Most youtubers shy away from the math, but not you. Your visual proofs are brilliant and will span through the ages, I thank you because I have genuinely been looking for this content for years. Out of the bottom of my heart thank you, I needed this...

[@akaisekai143]

Most Memorable: getting the Mathologer seal of approval

[@conoroneill8067]

The variations on the Harmonic series were definitely my favourite - who even thought to ask such a strange question as "What's the Harmonic Series, but if you remove all the terms with a nine in them?" It would never have occurred to me to ask a question like that.

[@TheGrenvil]

It's like the "Bee movie, but without bees" type of memes, I guess it's just the human nature

[@shaunsawyer8298]

So much great content packed into 45 minutes! Something I’ll always remember will be that the no nines series converges, and how simple the proof was!

[@randyhavard6084]

My new favorite maths channel

[@windturbine6796]

The most memorable part was me dying because I didn't know the no 9's series was convergent

[@MrTiti]

yeah, i just went for the odds and said: well it is finite, because maybe something will happen, that i cannot see now, so in contradiction i am still alive :) But... just to ask for clarity: if that grid exists with 9s, wouldt it exist with all other numbers >0 ? what about the 0 ?

[@jimmyh2137]

​@@MrTiti The grid is the "same" with every digit, including 0. In the first 10 numbers, 1/10 contain a (insert digit) or 10% In the first 100 numbers, 1/10 for every "ten" + 9/100 (the ones starting with your digit, like 31 32 33...), or 19% containing your digit. In the first 1000 numbers we have 19 every 100, + the ones with the first digit (300, 301, 302...), or 271/1000 or 27.1% total. And so on... Every time the % of numbers containing the chosen digit keeps increasing, reaching almost 100%. Works for all 10 digits (0, 1, 2... 9).

[@lc7269]

The most memorable was the optimal towers, as I always thought that the leaning tower of lire was the best way to stack overhangs. It looked so perfect that I never questioned if there was a better way to do it!

[@valeriobertoncello1809]

24:40 the sum of all the triangles that lower-approximate the blue areas is: 1/2*(1*(1 - 1/2) + 1*(1/2 - 1/3 )+ 1*(1/3 -... = = 1/2*(1 - 1/2 + 1/2 - 1/3 + 1/3 -... = 1/2*(1)= 1/2

[@vik24oct1991]

you have to also prove that the left over portion is greater than area of triangle.

[@sadkritx6200]

@@vik24oct1991 yes, that's why he said lower approximate. The left over portions are convex. So thinking of them as triangles, there's some area left. That's why the total area is greater than 1/2.

[@vik24oct1991]

@@sadkritx6200 That was my point , you don't need to calculate the sum of the area , if you prove that in each part the curves are convex then that implies that at the sum of the leftover is greater than half, no matter how the parts are divided.

[@davutkavranoglu6959]

Thank you very much for another excellent demonstration of the amazing beauty of mathematics! I love the 700 years old divergence proof. Also, the unbelievably slow pace of divergence is absolutely amazing.

[@PeterFreese]

Most memorable: the fact that gamma is the Ramanujan summation of the harmonic series.

[@typo691]

No way

[@AlfredJacobMohan]

You beat me to it.

[@PapaFlammy69]

Good Stuff Burkard! :)

[@noone7692]

POLSTER

[@jkid1134]

Oily Macaroni eh papa?

[@Ryan-gq2ji]

Mr flammable himself!

[@adityaruplaha]

Ayyy Papa

[@1976kanthi]

Papa Flammy!

[@royalninja2823]

A bit late on the lower bound for gamma, but... You can take every blue piece, place it into a rectangle of dimensions 1 x 1/(2^n), and split that rectangle in half with a diagonal from the top left to the bottom right. If you were to take the upper triangle from every one of these divided rectangles, you would get an area of one half of the square. Because every piece has a convex curve, it will stick slightly outside of the upper half of its rectangle. This means that every piece has an area greater than half of the rectangle, and the sum of all the pieces is greater than one half of the square. Because the square is 1x1, the area of the blue pieces is greater than 1/2.

[@Mathologer]

That's it. Never too late to have a great AHA moment :)

[@peteneville698]

@@Mathologer - Is it true to say that in the "No 'n's" series where we intuit that the sum converges, the sum of all the removed terms containing 'n' must itself be infinite? You have a series summing to infinity minus another series. If the thing you subtract is itself finite then you would still have an infinite series left over, ergo the subtracted series must itself sum to infinity for the remaining series to converge. Not sure if that's simply obvious or if it's also an "AHA" moment.

[@AndreaArturoGiuseppeGrossi]

I have to say, your channel is one of the best I am following about Math! Thank you very much for all these years of excellent work! :-D

[@gaelservignat2810]

The most memorable part was Tristan's fractal. Fractals are beautiful and they always show up when you expect them the least.

[@zswu31416]

"Is the no 9 series finite? You life depends on this!" Me: suspicious, has to be finite! "Believe it or not, it is finite!" Me: YAY

[@zswu31416]

PS this comment is a joke, I have heard about the no 9s series a long time ago

[@JM-us3fr]

Survival squad for the win!

[@shotgun3628]

@@JM-us3fr i would reply to you but ive been executed since i got it wrong

[@ryandvyn]

Most memorable: every positive number having its own infinite sum. It’s very obvious afterwards, but I would never believe it without your explanation. Thank you for all the interesting videos!

[@robertunderwood1011]

I love the fact that you give credit where credit is due to the lesser-known mathematicians whose work would be otherwise overlooked by most on this subject

[@robertunderwood1011]

You really give underlying thrill to the history and momentum mathematical exploration

[@gromburt]

Wonderful video as always. The more videos I watch the more I'm convinced that Euler must've been a time-travelling Mathologer viewer who really wanted to look smart by appearing in every video

[@techdecker51]

The most memorable for me has to be Kempner's proof, just due to how counterintuitive it is after seeing so many divergent series, but how intuitive the proof is.

[@GopikrishnaC-nj3sy]

24:30 (γ>1/2 :) It is equal to proving that the blue region is strictly greater than white region in that 1square unit box... since 1/x is concave up in (0,oo).. (Means a line formed by joining any two points on the curve (chord) will lie above the curve in that region) In those each small rectangles inside the 1unit box , the curve of each blue region (which is part of 1/x graph) will lie below the chord (here diagonol of that rectangle) As blue area crosses diagonals of each of these small rectangles (whose area is actually 1/(n) -1/(n+1) ) , it is greater than half the area of these rectangle... And adding up all thsoe rectangle gives area 1...and adding up all these small blue region is our "γ" So it is greater than half the area of 1. ie: γ>1/2. -----------------------------------------------

[@supercell6691]

Thanks

[@randombanana640]

basically their area is greater than their triangle counterparts and the triangle area is ½

[@leastsignificantbit5069]

Definitely the most impressive part is the animated Kempner's proof, I've expected something extremely complicated and yet the whole thing was "nice and smooth".

[@EntertainmentBooster]

His T-Shirt is always Unique... 👕

[@lebeccthecomputer6158]

I was about to comment on it; that’s an awesome shirt!

[@ahcuah9526]

And "infinitely" interesting.

[@kwinvdv]

And in the video also seem to show some kind of Moiré pattern behavior.

[@gcewing]

You can leave any time you like, but you'll never arrive...

[@AdityaSharma-qi8nu]

Most Memorable: The fact that it is possible to *arrange* the bricks on the table such that the last brick can be as far as the size of *observable universe* from the table, and yet be perfectly balanced!!!🤯🤯🤯

[@channalbert]

Pretty amazing, but don’t get the table from IKEA, they are bad quality.

[@AdityaSharma-qi8nu]

Lol

[@Fun_maths]

@@channalbert they mess with the physics of balance

[@Noname-67]

Imagine making a mistake and it all fall down

[@eefaaf]

Wouldn't that mean that you would need a stack higher than the universe? Even if each layer was just one atom thick.

[@Stev386]

Beautiful work you put into each and every episode!

[@mathwithjanine]

Wow this is so fascinating! Loved learning about all these amazing properties!

[@milan0234]

Most memorable: The proof that the bishop came up with, beautiful simplicity

[@zanedobler]

For me, the most memorable part was the optimal setup for 20 bricks because it made me glad I'm not an architectural engineer.

[@cheyennesadeghi]

My favorite part was when you revealed that the sum of the 100 zeroes series is greater than the sum of the no 9 series. Absolutely mind-blowing. In truth, my favorite part was the entire video you just made me pick :) Thank you!!!

[@SeanBask]

Most memorable part: the visualization of the "no nines sum convergence" What an awesome way to look at it.

[@mitchkovacs1396]

24:35 You can construct a right triangles out of the corners of each blue region. The base of each is 1 unit while the height is 1/n - 1/(n+1). The sum of the areas of these triangles yields a lower bound for γ. We can see that this area is (1/2)*(1 - 1/2) + (1/2)*(1/2 - 1/3) + (1/2)*(1/3 - 1/4) + ... which is a telescoping series so we can cancel everything except 1/2*1, so 1/2 is a lower bound for γ

[@Mathologer]

That's it. Of course, you can also just skip the algebra :) Having said that it's nice in itself that all this corresponds to a telescoping sum when you turn it into algebra.

[@levmarchuk998]

Most memorable: fractal visualization of no-9's series being finite

[@pixels_]

Agreed! Simple and thus memorable

[@doctortrouserpants1387]

Thanks for another amazing, informative, extremely clear and well made presentation. The fact that the series explodes to infinity is one thing, how slow it happens and how many terms are needed for just a tiny increase makes my head spin.

[@diadochokinetic3290]

Great stuff. It's always amazing how you manage to find such intuitive explanations. Most memorable is probably the "no 9s" visualization.

[@SzanyiAtti]

Exactly what I needed today EDIT: My favourite part was the no 9-s proof. It is just simply elegant.

[@elainechegwinhall52]

The most memorable is the fact of the partial sums being non integer.

[@onomatopoesi]

I agree, that is really fascinating.

[@rubyjohn]

Awesome video! Your excitement is truly contagious through the video :D

[@christosgolsouzidis1286]

The most memorable part is the connection of the 'γ' and the log() function to the harmonic series! Really amazing!!

[@pedrobress1072]

My vote definetely goes to Kempner's proof, it is extremely elegant, since the concepts used are individualy simple, such as the calculation of numbers without nine or geometric series, but when cleverly combined they form this amazing result. Besides that, great video as always. Edit: typo

[@cauankazama4431]

I agree with you!

[@michaellucas7177]

Marble: Most memorable Idea: No integers among the partial sums. Your excitement is always a great feature not your presentations. And why you're my favorite Mathematics KZbinr!

[@BjornAlexanderMay]

The most impressive part in my opinion was the fact that the 100 zeros sequence converges to a bigger sum than the no 9s sequence. Greetings from Germany by the way and keep up that great work. It is always a pleasure diving into your mathematical discoveries!

[@ngbusca]

This channel is such a joy! Loved the leaning tower of Lira :-)

[@inakidesantos9209]

Most memorable: The 700 year old proof by a bishop

[@davidrheault7896]

11 views , let's make it prime ;p I also enjoyed the 700 year-old proof

[@jksmusicstudio1439]

First time I saw that proof I was like "oh come on, that's pure hacking" XD

[@Perririri]

A rook is still better, though! 😂

[@maxinator2002]

Most memorable: the optimized leaning tower! Although it was very messy, I think there's a lot of beauty in the fact that the most optimal arrangement of bricks is such a mess. It reminds me of how an extremely simple physical system (like a double pendulum) can result in chaos!

[@oliverave1234]

Good point. The problem with I have that solution is perhaps that the original concept stipulated or assumed that the tower would lean only in one direction (which is what towers do). The recent innovative solution doesn't involve a "tower" at all. This is the problem with the "thinking outside the box" cliche. I tell you to connect the dots within a box using 2 lines and then you connect them by drawing lines outside the box. Wow, that's impressive. It's like telling your boss you can solve a problem by spending 10X the given budget.

[@aleksitjvladica.]

I love you! Always smiling, wonderful and clean animation and explanation. Such composition inspires a person to work on mathematics immediately. Much of the work and love you put there. And My new co-worker looks so much like you. Also, balanced warm-up shocked Me, I am so excited to think about it and to continue the video.

[@Omni-Kriss]

I love the little chuckles you have, your joy is contagious :)

[@Richardstureborg]

solution to there bricks with overhang of 2 units: place one brick with overhang of 1. then place another bring on top of this one all the way to the right with its own overhang of 1 unit. clearly this will fall. now place the third brick to the left of the second, making the top layer 4 units (2 bricks) long, and the bottom layer centered around it. Done.

[@canaDavid1]

Summary: Cliff edge is x=0. Bricks are measured at the middle. Layer 0: a brick at x=0 Layer 1: two bricks, at -1 and 1

[@l3p3]

Idea: Put the left upper coin before the overhanging one. It will not fall.

[@pummelfee9093]

Wow, ich hätte nie geglaubt, dass etwas das mit Analysis zu tun hat auch Spaß machen kann... :) Sehr cool!

[@urmi-e-azar1776]

The curious convergent series made my day! Baillie's paper was amazing. I'm now going to look at your presentation of Kempner's proof

[@idalinofranco1121]

This is truly amazing and inspiring! Such an old series and the physical properties it can describe are still mostly "under cover"

[@MohammedAbdullah-mx1vg]

I have to vote for the Kempner's proof animation, it was simply stunning to see such a seemingly complex problem; being broken down into techniques that a school student could understand 👏

[@RichardQuadling]

I wish I knew more. But the more I watch and try to learn, the more time I've used up getting nowhere. The dedication the geniuses have to mathematics and physics is astounding. And it is not done for reward other than the pursuit of knowledge. And that is as beautiful as the proofs the geniuses present. Thank you for the videos.

[@derrikzimmermann9491]

I love your video. I have a minor in Math and never had a prof explain visually any of these concepts like you have. Thank you it has revitalized my love for math!

[@santiagolasso141]

I would have never related the armonic series with a leaning tower, amazing!

[@citizenscientistsworkshop1948]

Most memorable: Tristin’s visual proof of the finiteness of the no-9 series.

[@johncanfield1177]

Dear Mathologer, I am so pleased whenever I run across one of your videos. As for my vote for the portion that impressed me the most, it would have to be the leaning tower of Lire. There is something so lovely in its orderliness, that I sense my head bowing, much like the old Frenchman, Oresme. Thank you for another interesting and entertaining video on the beauties of math.

[@jeewanjotsingh3088]

This was probably one of the best video that I have seen on this topic. And the geometric limit of Gamma between 1/2 and 1 and the no-nine series proof was something I learnt for the first time.

[@bryanbischof4351]

When he put the stack leaning over Oresme I lost it.

[@Mathologer]

had to be done :)

[@damaoguan765]

Most memorable: Thought that I knew what was a harmonic series until watching your video! Now I know something about what it can be and what part of it can be.

[@ValkyRiver]

Me: microtonal music.

[@ludovicrivet6249]

My favorite fact definitely was how the no partial sum of the harmonic series adds to an integer. I am a big fan of these little tricks that doesn’t look very interesting at first glance, but are just absolutely delightful. Once again, than you for an amazing video!

[@andrewsteel5611]

Its pretty amazing that you have all these finite series that together make the infinite series - i liked the part when you posed the question about the no nines

[@angstony459]

Most memorable: Chapter 1: "Let's assume that the grey bar does not weigh anything - thought experiment - we can do this - hehe" Top notch video!

[@davidrheault7896]

It (the likes) was prime, I clicked and it remained prime ;p