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

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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 RU-vid 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)

Опубликовано:

9 июн 2024

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Комментарии : 3,5 тыс.

@daemos_magen 3 года назад

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 3 года назад

😂😂😂

@douglasstrother6584 3 года назад

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

@christiaanbalke 3 года назад

Same over here :)

@teleny2 3 года назад

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

@douglasstrother6584 3 года назад

@@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."

@bernyelpro1906 3 года назад

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

@christianorlandosilvaforer3451 3 года назад

sure.. x2

@Torthrodhel 2 года назад

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 2 года назад

@@Torthrodhel t

@sergiomv85 2 года назад

I lost my life too!

@apbmes7690 3 года назад

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 2 года назад

We've all had those graders.

@snowstarsparkle 2 года назад

@stopthephilosophicalzombie9017 2 года назад

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 2 года назад

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

@Meta7 2 года назад

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

@n0nam3given 3 года назад

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 3 года назад

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

@jzieba0204 3 года назад

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 8 месяцев назад

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.

@Fun_maths 3 года назад

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

@Mathologer 3 года назад

@johnchessant3012 3 года назад

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 3 года назад

@@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 3 года назад

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

@Fun_maths 3 года назад

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

@sergeboisse 3 года назад

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

@Achrononmaster Год назад

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 Год назад

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

@manelmanolo7195 Год назад

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

@TravisTellsTruths Год назад

Exactly true 👍

@whatby101 3 года назад

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.

@shivambiswas5790 3 года назад

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

@456MrPeople 3 года назад

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 3 года назад

@@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 3 года назад

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

@sharpfang 3 года назад

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

@EdoTimmermans 3 года назад

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

@Meepmows 3 года назад

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 3 года назад

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 3 года назад

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

@dennisbell9639 3 года назад

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

@mattbox87 3 года назад

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 2 года назад

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

@PC_Simo 14 дней назад

My thoughts, exactly 🎯! Articulated better, than I could have put it 😌👍🏻.

@zacharystark5520 3 года назад

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

@TheM0JEC 3 года назад

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

@MasterHigure 3 года назад

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 3 года назад

@@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 3 года назад

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

@parkershaw8529 3 года назад

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

@karateoscar 3 года назад

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

@Jack_Callcott_AU 3 года назад

That got me too.

@thatssomethingthathappened9823 3 года назад

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

@PC_Simo 15 дней назад

That’s my favourite / most memorable part, too; because I managed to prove it, for myself, and find the pattern, for the nth partial sum: Σ(n) = (a(n-1)*n+(n-1)!)/n!, where a(n-1) = the numerator of the (n-1)th partial sum. 🙂

@danield1303 3 года назад

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

@I_like_math 3 года назад

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

@jean-francoistremblay7744 3 года назад

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.

@dEntz88 3 года назад

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 3 года назад

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

@dEntz88 3 года назад

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

@morodochable 3 года назад

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.

@dragifire 3 года назад

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!

@bagochips1208 3 года назад

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

@sadkritx6200 3 года назад

Same bro. The whole video was magnificent 👌

@anon6514 3 года назад

They should have called it the 'Barely Divergent Series'

@takyc7883 3 года назад

is t the slowest diverging series?

@yazeed0ps3 3 года назад

@@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 3 года назад

@@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).

@_kapy_ 3 года назад

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

@Sn0wjunk1e 3 года назад

you mean beautiful

@EdoTimmermans 3 года назад

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

@tim40gabby25 3 года назад

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 3 года назад

It was the most beautiful leaning tower.

@ammaleslie509 2 года назад

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"

@aegirson2899 3 года назад

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

@naimulhaq9626 3 года назад

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

@rohitjoshi2953 3 года назад

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.

@thek3nger 3 года назад

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 3 года назад

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

@jackw7714 3 года назад

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

@stephenmcateer 3 года назад

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 3 года назад

how was the paper?

@gregburnell8454 3 года назад

This blew my mind!

@plienair 3 года назад

Most memorable moment was the cat going "μ".

@teleny2 3 года назад

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

@PC_Simo 14 дней назад

As a cat-purrson, I approve 😻😌👍🏻.

@valeriobertoncello1809 3 года назад

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 3 года назад

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

@sadkritx6200 3 года назад

@@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 3 года назад

@@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.

@eruwa1550 3 года назад

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

@justsomeguy5628 3 года назад

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

@Igneous01 3 года назад

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 3 года назад

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 3 года назад

@@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 3 года назад

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

@ramenandvitamins 3 года назад

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

@peter_p_r_zhang 3 года назад

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 3 года назад

That's it:)

@youssefm1 2 года назад

Why is the total area of those triangles 0.5?

@Ohhelmno 2 года назад

@@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 2 года назад

@@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 2 года назад

Don

@davutkavranoglu6959 3 года назад

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.

@GopikrishnaC-nj3sy 3 года назад

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 3 года назад

Thanks

@randombanana640 2 года назад

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

@alexgreer878 3 года назад

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

@davidgustavsson4000 3 года назад

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

@amadeus1940 3 года назад

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

@ummerfarooq5383 3 года назад

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 3 года назад

I relate to that experience.

@rexwhitehead8346 3 года назад

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.

@cheyennesadeghi 3 года назад

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 3 года назад

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

@johnchessant3012 3 года назад

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 3 года назад

Exactly :)

@MonsieurBiga 3 года назад

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

@anthonycousins853 3 года назад

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

@jisyang8781 3 года назад

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

@moritzalshuth7239 3 года назад

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

@lc7269 3 года назад

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!

@royalninja2823 3 года назад

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 3 года назад

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

@peteneville698 2 года назад

@@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.

@PC_Simo 14 дней назад

That is very much true 🎯👍🏻.

@danicorrales1592 3 года назад

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

@conoroneill8067 3 года назад

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 3 года назад

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

@neomew 3 года назад

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

@ashtonsmith1730 3 года назад

so how did math work before euler?

@neomew 3 года назад

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

@raphaelreichmannrolim25 3 года назад

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

@andrewxc1335 3 года назад

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

@davidrheault7896 3 года назад

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

@leastsignificantbit5069 3 года назад

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".

@HyperFocusMarshmallow 2 года назад

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

@gaelservignat2810 3 года назад

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

@akaisekai143 3 года назад

Most Memorable: getting the Mathologer seal of approval

@takeguess Год назад

I have to say you are one of my favorite RU-vidrs! 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...

@ryandvyn 3 года назад

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!

@gromburt 3 года назад

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

@PeterFreese 3 года назад

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

@typo691 3 года назад

No way

@AlfredJacobMohan 3 года назад

You beat me to it.

@christosgolsouzidis1286 3 года назад

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

3 года назад

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!

@techdecker51 3 года назад

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.

@mitchkovacs1396 3 года назад

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 3 года назад

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.

@elasiduo108 3 года назад

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

@randyhavard6084 2 года назад

My new favorite maths channel

@windturbine6796 3 года назад

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

@MrTiti 3 года назад

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 3 года назад

@@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).

@zswu31416 3 года назад

"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 3 года назад

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

@JM-us3fr 3 года назад

Survival squad for the win!

@shotgun3628 3 года назад

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

@AndrewMooreMar 3 года назад

The e^gamma equation really blew my mind. Good stuff!

@citizenscientistsworkshop1948 3 года назад

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

@PapaFlammy69 3 года назад

Good Stuff Burkard! :)

@noone7692 3 года назад

POLSTER

@jkid1134 3 года назад

Oily Macaroni eh papa?

@Ryan-gq2ji 3 года назад

Mr flammable himself!

@adityaruplaha 3 года назад

Ayyy Papa

@1976kanthi 3 года назад

Papa Flammy!

@alejandronasifsalum8201 3 года назад

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!

@sebastiansimon7557 3 года назад

I liked the crazy optimal overhang tower the most. Didn’t expect that at all.

@richardatkinson4710 Год назад

Me too. The untidy structure called to mind Sabine Hossenfelder’s “Lost in Math: How Beauty Leads Physics Astray”. Physicists are addicted to symmetries. Dirac thought beauty was the most important (and convincing) feature of any formula in mathematical physics.

@ivansmitt2195 3 года назад

as for me its such a charm to see the Mathologer seal of approval, I really dont know, but it really made my day!! thanks for the great video!

@maxinator2002 3 года назад

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 2 года назад

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.

@michaellucas7177 3 года назад

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 RU-vidr!

@shaunsawyer8298 3 года назад

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!

@AndreaArturoGiuseppeGrossi 3 года назад

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

@pedrobress1072 3 года назад

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 3 года назад

I agree with you!

@dhruvsingh34 3 года назад

His T-Shirt is always Unique... 👕

@lebeccthecomputer6158 3 года назад

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

@ahcuah9526 3 года назад

And "infinitely" interesting.

@kwinvdv 3 года назад

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

@gcewing 3 года назад

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

@tammylynn8632 3 года назад

The most memorable is your voice... the giggle you make when telling us wonderful facts. Have a wonderful life. Stay safe... ✌️

@trueriver1950 3 года назад

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

@levmarchuk998 3 года назад

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

@pixels_ 3 года назад

Agreed! Simple and thus memorable

@SzanyiAtti 3 года назад

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

@jacoboppenheim2385 3 года назад

Most Memorable: Kempner's proof at the end. Remarkable video!

@diadochokinetic3290 3 года назад

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

@Richardstureborg 3 года назад

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 3 года назад

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 3 года назад

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

@inakidesantos9209 3 года назад

Most memorable: The 700 year old proof by a bishop

@davidrheault7896 3 года назад

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

@jksmusicstudio1439 2 года назад

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

@Perririri 2 года назад

A rook is still better, though! 😂

@Skunz02 3 года назад

The most difficult question to answer is what part to vote for. If my life depended on it, I would pick the “no integers in the partial sums” topic ... and surrender my life for stealing gamma. Awesome video, wonderfully addictive, as usual!

@jacobparish8134 3 года назад

Most memorable: that cool proof that gamma < 1 by sliding all the blue regions into the unit square

@davidrheault7896 3 года назад

Of course I have done a proof and a computation of the digamma function and the Euler's constant gamma. For low values of n (harmonic series partial sum), use the Bernoulli numbers in association with gamma and you get the true value to an amazing precision

@milan0234 3 года назад

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

@AdityaSharma-qi8nu 3 года назад

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 3 года назад

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

@AdityaSharma-qi8nu 3 года назад

Lol

@Fun_maths 3 года назад

@@channalbert they mess with the physics of balance

@Noname-67 3 года назад

Imagine making a mistake and it all fall down

@eefaaf 3 года назад

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

@doctortrouserpants1387 2 года назад

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.

@neilaryaofficial 9 месяцев назад

Love your work, thank you for sharing so much knowledge & quality content. Generally RU-vidrs run for views, nothing more than that. You never click baits or anything that ever had made me regret clicking. You are just awesome.

@pummelfee9093 3 года назад

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

@zanedobler 3 года назад

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

@adriannillsen8680 3 года назад

It is really interesting how eulars comes into all these different formulas. Personal favourite was the very neat proof at the end that the sum was less than 80

@johncanfield1177 2 года назад

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.

@MohammedAbdullah-mx1vg 3 года назад

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 👏

@prabesh321 3 года назад

Most memorable: Sad looking portrait of Nicole Oresme along with the Leaning Tower of Lire

@anselmobd 3 года назад

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

@etiennnelacroix4653 2 года назад

Danke für diese wundervollen Videos!

@elainechegwinhall52 3 года назад

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

@onomatopoesi 3 года назад

I agree, that is really fascinating.

@bryanbischof4351 3 года назад

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

@Mathologer 3 года назад

had to be done :)

@SlifeSify 3 года назад

Thank you for making math visually interesting.

@ArchimedesBC 3 года назад

Mathologer: I love your geometrical intuition, great job!

@angstony459 3 года назад

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 3 года назад

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

@robertunderwood1011 2 года назад

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 2 года назад

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

@jeewanjotsingh3088 3 года назад

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.