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.
@@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."
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.
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.
@@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.
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.
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
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.
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
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? 👋🕊️
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.
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
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.
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.
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.
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
@@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.
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!!!
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
I have to say you are one of my favorite RUclipsrs! 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...
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!
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 - 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.
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.
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.
Most awesome unthinkable thing for me was that the sum crosses all these infinite integers at such a slow pace and yet doesn't touch any!! BLEW MY MIND!!!
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
Mathloger THE VERY BEST MATH AND LOGIC TEACHER ON THE PLANET. I missed him already when I was a school boy without knowing him. I enjoy every clip and its so well presented and easy to understand. He*'s a NATURAL AND A PRO... go on bro
The most memorable is definitely Kempner's proof. Seeing it, I thought to myself "damn, I should have seen this coming!". PS. if I get lucky, please don't send me "The Mathematics of Juggling", I already have it :)
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!
For one brick to be completly beyond the edge using only three bricks, an upside down pyramid centered right at the edge would work ! Astounding work btw :)
First off, let me tell you how great these animations are. For my calculus students, I will definitely show them the part with the leaning tower of Lira. I also enjoyed the 700 year-old proof from a Bishop. I knew the integral test proof which I usually do in class. Of course, if we replace every other term by a minus sign, then the series partial sum converges to ln(2).
The fact that all the partial sums are nonintegral was quite memorable; especially once it was explained the graphic at the beginning of the chapter made so much more sense. Also the fact that the sum of the “100 zeros” series is _greater_ than the sum of the “no 9s” series (assuming the approximations were of similar accuracy) despite being less dense is quite mind boggling - that one I’ll certainly not forget 🤯 Edit: skimmed the paper and I believe the term these days is “nice”
@@Mathologer Yeah, me too. I was completely ready to vote for the gamma section, but then that proof blew me away (because I could see the end coming).
@@lynnwilliam depends on the contry and the university you are in. In my uni we see differential geometry, functional analysis, partial differential equations and galois/field theory.
Why the divergent harmonic series cannot proof Zeno Paradox of movement ? Why is proved with the series of 1/2^n and not with that of 1/n if both sequences tends to 0 ? Ok series 1/2^n is convergent and our harmonic divergent, but if Zeno asked for 1/2, 1/3 ,1/4, 1/5 ,...,1/n to the destination, this divergent is one case Against the other convergent used to proof of movement. Is not a "math cheat" choose one sequence convenient to the proof without explain why others are not valid AGAINST that choosen? If the answer is only "divergent is infinite sum" this gives reason to Zeno, as he have one valid sequence against that convergent proof. Who wins? I think have answer to this, but post my initial fair though.
It's your short mention at 28:33 of that part that I've seen and heard so many times before, that's been haunting math videos for years! You know, the Ramanujan sum thing, that the sum of all positive integers is finite, negative, and a fraction! You actually SAY it's not equal, but I've been told by at least one RUclips commenter that it is. I can't help but wonder if Ramanujan, Hardy and Littlewood are all laughing in their graves.
I literally just came from a video of Douglas Hofstadter talking about his INT function, and now there's 46 minutes of Mathologer right after? Better take a break before I take this in. :)
@@Mathologer it's /watch?v=W70-LCoAYuQ . I found it fascinating how you can go from counting the triangle numbers between squares to a fractal function that's discontinuous at every rational point!
The fact that sum of reciprocal of numbers having googol zeros is larger than the sum of no 9 series is really mind bending. Also the music in credits is quite nice. :)
Love this! Favourite part by far the fact that these fractions turn out odd over even always so the sum never ever hit an integer? to infinity! That’s just nuts
Thanks for a fun math video to finish off the week. #my_favorite_fact was that there are no integers among partial sums (excluding 1). This and Euler's gamma number can really mess with people's OCD.
Most memorable part: is that as a middle school boy I was obsessed with infinite sums, some of which are in this video, I never knew how important they were.
Most memorable part was looking at the blue curves inside the 1x1 block and thinking "well sure it's less than one, but is it really negligible? It's still more than a half" and then you asked us to state why it's obvious that it's more than one half.
Most memorable moment? For me, it had to be noting the stack of infinitely many "blue bits" would fit inside the unit square before you mentioned it! As for the proof of gamma > 1/2, each slice of that stack would equal one half if the curved part of the shape were in fact a straight diagonal; since the curve is convex with respect to the "blue bit," the shape's area is greater than half of the rectangle it occupies. It's really gratifying to be on top of a concept in a Mathologer video, considering that in a typical video (including this one!) there are usually parts that I have to go back and rewatch to make sure I understood correctly! 🤣
"While you often encounter beauty in the optimal, this is definitely not always the case." at 10:01. And then I think of people, personalities. An interesting truth.
Never an an integer as the value for any partial sum - that is mind-blowing, since the series runs forever and there are an infinite number of candidate integers along the way, yet the sum misses hitting any integer spot on. It is easier to manage mentally when I recall that there are an infinite number of decimal places demanding all zeroes for a number to call itself an integer.
I really, really, really enjoyed watching this video. Your channel is such a great example for how intriguing and fun mathematics can (and should) be. Even though I can't really pin down my favourite chapter of the video, for me personally, the results by Robert Bailie as well as the fact that the harmonic series' Ramanujan summation is the Euler-Mascheroni constant were the most interesting. Here in Austria, to get a high school diploma, one must write a ‘Vorwissenschaftliche Arbeit’ (basically a ~25-page thesis about your topic of choice). I wrote about the connection between primes and the Riemann hypothesis and did a lot of research about the harmonic series. I already knew the proof by Oresme, but the rest of the video was mostly completely new to me and I am so thankful for this video. Liebe Grüße aus Österreich :)
You asked which part impresses me most. Well, your enthusiasm impresses me most. I am too old to remember how to prove the divergence of the harmonic series, yet I enjoy everything you mentioned in the video. [Don't bother to send me any of your books. I won't be able to finish reading any of them. :-)
The most memorable part was the leaning tower of lira for two reasons: first, it was a fantastic exercise to have high school students actually perform; second, because it was "coined" by Paul Johnson. Third time I showed this video I realized what was said and started laughing, then had to explain myself.
I liked the part in which the series is approximated to ln(n+1)+gamma. And the fact that the more you go on, the better this approximation is valid. Which, as you said, quite interesting! Usually, the further you go, the worse it gets but not in this case. Very interesting! EDIT: the demonstration of the no 9s sum is super cool!
Most memorable: finding out the various no 1's, no 2's, no 3's ... no 9's infinite sums are all finite and that you can find an approximate value for the sum of the Harmonic series with a simple formula.
Most memorable part: the connection between the leaning tower problem and the harmonic series. Had already seen an MIT demo of that, but had never attempted to do any calculations
I liked most the approximation of the harmonic series with the ln-function, and especially the easy geometric argument, that gamma is between 0.5 and 1.
To have one block beyond the clif line with three blocks, the base needs one with the center mass on the line and the second "floor" two with its center of masses on the extremities of the one on the base
The most memorable part for me was to show that the error between ln(n+1) and the n-th partial sum of the harmonic series was smaller than one. Using it you don't only get a good approximation but also show that the harmonic series doesn't converge.
I'm so happy to see this exercise here. This is an exercise I had when I was in Montpellier Université II Faculty of Science (France) in Mathematics Studies. And as far as I know, I was the only one to have to idea to start from the top. So the only one I know out of the full amphitheatre) who did the exercise. Well the rest was not easy we had to tell if it's gonna go infinite or not at the limit. I had to work hard to put it on paper as it was obvious but hard to prove. (For me at least). But the very interesting fact about this exercise, is, it is THE exercise that made me stop studies. Doing this exercise I dicovered something else, that was of a much higher value.
Watching this video inspired me to make a discovery. You can approximate e^x using the harmonic series. Use the series of integers S[x] defined as follows: start summing the harmonic series until the sum is first >= x. The number of HS terms in the sum is the first term of the series, i.e. S[x](1). Then starting with the next number of the harmonic series, rinse and repeat until you have the number of terms you desire. The ratios S[x](n+1)/S[x](n) form increasingly better rational approximations of e^x as n gets larger. This even works for 0 since S[0] = 1, 1, 1, ... and e^0 = 1/1. And it can be made to work for negative x by using the series for abs(x) and inverting. It can't be the case that I'm the first to notice this, so who is credited with this historically?
For me favourite thing about this video was realizing how the armonic series never lands on an integer. I remember trying some finite sums of the reciprocals with my calculator, when they told us that the complete series was divergent. And now watching this I was amazed by the simple of the argument on how the partial sums always fall outside of the natural numbers.
The implication of the cantilever block towers is that an overhang of ANY amount can be achieved given enough blocks. So my immediate questions would be 1) How many blocks are minimally necessary to achieve a particular overhang, 2) What is the maximum number of whole blocks that can be positioned beyond the edge of the cliff for a balanced tower of a) a certain number of blocks, b) a certain height. My favorite part is the algebraic proof at the end, and that one of the contributors names is “Ooo”!
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.
😂😂😂
When she hears all of the profanity, she knows you're watching Flammable Maths!
Same over here :)
His giggling always sounds like Dr. Strangelove to me. Man, Peter Sellers was a great actor.
@@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."
Mathologer video series are definitely better than any Netflix series. They surprise me anytime.
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.
Netflix? No comparison. Mathologer wins every time, and it's free.
Mathflix. The best series (Taylor, MacLaurin, armonic, ...)
(Seen in his t-shirt)
Exactly true 👍
Most memorable part: me losing my life after failing the “no nines sum converges”
sure.. x2
I didn't lose my life at that part! I gamed the system, by already losing it way earlier on in the video! lmao
@@Torthrodhel t
I lost my life too!
If you had collected 1000 Coins (or more), in Japan; then, it’s no problem. 🙂
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.
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.
It's like the "Bee movie, but without bees" type of memes, I guess it's just the human nature
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.
Exactly how I pictured it!
It also makes it obvious that γ is much closer to ½ than it is to 1.
Fred
My thoughts, exactly 🎯! Articulated better, than I could have put it 😌👍🏻.
Most memorable: that the harmonic series narrowly misses all integers by ever shrinking margins
I agree that an infinite number of non intergers is quite amazing.
I mean, any diverging series with ever smaller terms will have ever shrinking margins (as long as it doesn't actually hit any integers).
@@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.
@@landsgevaer You're right. The terms need to converge to 0. I done goofed.
It also managed to miss infinitely more and infinitely denser all irrational numbers as well. THAT seems even more impressive!
The most memorable part was Tristan's fractal. Fractals are beautiful and they always show up when you expect them the least.
For me, the "no integers" part was the most memorable, but honestly the whole video was of great quality (as expected).
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.
Exactly :)
Finally something I had seen myself with my very low level of maths
That makes sense! Good explanation, I got it without any visuals! Haha.
You mean convex. You triggered one of my pet peeves.
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
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.
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
Most memorable: The most efficient overhanging structure being the weird configuration instead of an apparently more ordered one.
Most memorable: An overhanging structure with n=google bricks.
I liked your evil mathematician back story, with the teacher refusing to grade the "wrong" proof.
I didn't like it. Second hand annoyance. grrr
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?
👋🕊️
I relate to that experience.
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.
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
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.
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.
The weirdest thing you showed is definitely the unusual optimal brick stacking pattern.
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.
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
@@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.
@@ramenandvitamins sorry, typo from my side.I meant top left, those 3 that are stacked exactly over each other
@@maze7474 I suspect they'd no longer suffice to hold down the second-rightmost block if they were moved any further left.
Most memorable moment was the cat going "μ".
Has a cat the hacker-nature? "Mew...."
As a cat-purrson, I approve 😻😌👍🏻.
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!!!
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
I agree with you!
Most Memorable: getting the Mathologer seal of approval
The most memorable part is the connection of the 'γ' and the log() function to the harmonic series! Really amazing!!
Good Stuff Burkard! :)
POLSTER
Oily Macaroni eh papa?
Mr flammable himself!
Ayyy Papa
Papa Flammy!
Most memorable: If my life depended on knowing if the sum of no nines series is finite I would not be alive
I have to say you are one of my favorite RUclipsrs! 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...
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!
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!
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.
That's it. Never too late to have a great AHA moment :)
@@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.
That is very much true 🎯👍🏻.
You’re so good at starting simple and yet including stuff that’s interesting for the fully initiated! Great work!
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.
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
Idea: Put the left upper coin before the overhanging one. It will not fall.
The most memorable is your voice... the giggle you make when telling us wonderful facts. Have a wonderful life. Stay safe... ✌️
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!
It (the likes) was prime, I clicked and it remained prime ;p
For me, the most memorable part was the optimal setup for 20 bricks because it made me glad I'm not an architectural engineer.
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.
His T-Shirt is always Unique... 👕
I was about to comment on it; that’s an awesome shirt!
And "infinitely" interesting.
And in the video also seem to show some kind of Moiré pattern behavior.
You can leave any time you like, but you'll never arrive...
Most memorable: Sad looking portrait of Nicole Oresme along with the Leaning Tower of Lire
Most awesome unthinkable thing for me was that the sum crosses all these infinite integers at such a slow pace and yet doesn't touch any!! BLEW MY MIND!!!
Most memorable: Tristin’s visual proof of the finiteness of the no-9 series.
Great stuff. It's always amazing how you manage to find such intuitive explanations. Most memorable is probably the "no 9s" visualization.
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
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
Mathloger THE VERY BEST MATH AND LOGIC TEACHER ON THE PLANET. I missed him already when I was a school boy without knowing him. I enjoy every clip and its so well presented and easy to understand. He*'s a NATURAL AND A PRO... go on bro
I would have never related the armonic series with a leaning tower, amazing!
Most memorable: Kempner's Proof. Just pipping the 100 zeros weirdness.
I explode into laughter several times, watching your videos, you have a superb sense of humor.
When he put the stack leaning over Oresme I lost it.
had to be done :)
The most memorable is definitely Kempner's proof. Seeing it, I thought to myself "damn, I should have seen this coming!".
PS. if I get lucky, please don't send me "The Mathematics of Juggling", I already have it :)
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!
I think what I liked most was the proof that the series diverges. It's very clever!
For one brick to be completly beyond the edge using only three bricks, an upside down pyramid centered right at the edge would work ! Astounding work btw :)
First off, let me tell you how great these animations are. For my calculus students, I will definitely show them the part with the leaning tower of Lira. I also enjoyed the 700 year-old proof from a Bishop. I knew the integral test proof which I usually do in class.
Of course, if we replace every other term by a minus sign, then the series partial sum converges to ln(2).
The most interesting part to me was that there are no integers on the way to infinity for the harmonic series. Thanks for the video. It was excellent.
The fact that all the partial sums are nonintegral was quite memorable; especially once it was explained the graphic at the beginning of the chapter made so much more sense. Also the fact that the sum of the “100 zeros” series is _greater_ than the sum of the “no 9s” series (assuming the approximations were of similar accuracy) despite being less dense is quite mind boggling - that one I’ll certainly not forget 🤯
Edit: skimmed the paper and I believe the term these days is “nice”
The people who dislike this kind of videos are really foolish. I am big fan of you sir. I love the way you explain something.
There are a LOT of very angry and nasty and foolish people out there :(
The most memorable is Kempners proof!
Two votes for Kempner so far. Very interesting :)
@@Mathologer Yeah, me too. I was completely ready to vote for the gamma section, but then that proof blew me away (because I could see the end coming).
this is a great topic for an undergrad first year calculus class.
I teach some of this stuff every year at uni. Will also be nice to have this video to show to my students :)
If this is 1st year at Uni. What is year 4? Wow
@@lynnwilliam depends on the contry and the university you are in. In my uni we see differential geometry, functional analysis, partial differential equations and galois/field theory.
Why the divergent harmonic series cannot proof Zeno Paradox of movement ? Why is proved with the series of 1/2^n and not with that of 1/n if both sequences tends to 0 ? Ok series 1/2^n is convergent and our harmonic divergent, but if Zeno asked for 1/2, 1/3 ,1/4, 1/5 ,...,1/n to the destination, this divergent is one case Against the other convergent used to proof of movement. Is not a "math cheat" choose one sequence convenient to the proof without explain why others are not valid AGAINST that choosen? If the answer is only "divergent is infinite sum" this gives reason to Zeno, as he have one valid sequence against that convergent proof. Who wins? I think have answer to this, but post my initial fair though.
28:50 Gotta love the Numberphile-burn 😅.
It's your short mention at 28:33 of that part that I've seen and heard so many times before, that's been haunting math videos for years! You know, the Ramanujan sum thing, that the sum of all positive integers is finite, negative, and a fraction! You actually SAY it's not equal, but I've been told by at least one RUclips commenter that it is. I can't help but wonder if Ramanujan, Hardy and Littlewood are all laughing in their graves.
Burkhard you are awesome I have read Havil's book and you are an incredibly valuable resource. Thank you
Wow this is so fascinating! Loved learning about all these amazing properties!
I literally just came from a video of Douglas Hofstadter talking about his INT function, and now there's 46 minutes of Mathologer right after? Better take a break before I take this in. :)
Eddie Woo uploaded 20 hours ago. 🕰️
What video is that, maybe I should have a look myself :)
@@Mathologer it's /watch?v=W70-LCoAYuQ . I found it fascinating how you can go from counting the triangle numbers between squares to a fractal function that's discontinuous at every rational point!
@@Mathologer ruclips.net/video/W70-LCoAYuQ/видео.html
Here you go mate :)
@@mebamme Thanks for that, added to my Watch Later. I mean, I'm gonna watch it but there's no way I'm interrupting Mathologer.
Amazing! Just amazing how you can develop complex from a simple just through intuition. I wish schools also focused more on intuition just like you
The fact that sum of reciprocal of numbers having googol zeros is larger than the sum of no 9 series is really mind bending. Also the music in credits is quite nice. :)
You have great math, but often you get beyond me. Love your stuff!
Most memorable to me was that the sum of 100 zeros > no 9s sum. Just goes against every intuition I would have. Thanks for the video!
I love the optimal 20-block stack and find it entirely fascinating, not ugly in the least.
I love the little chuckles you have, your joy is contagious :)
Love this! Favourite part by far the fact that these fractions turn out odd over even always so the sum never ever hit an integer? to infinity! That’s just nuts
Fascinating. Thanks for posting this.
I lost it long before the "no 9's series" question......but you are a marvelous teacher!!!
Hooray!
Excellent Job.
Thank You
CAB
Thanks for a fun math video to finish off the week.
#my_favorite_fact was that there are no integers among partial sums (excluding 1).
This and Euler's gamma number can really mess with people's OCD.
Most memorable part to me is how simple the argument for the "no 9s" sum was in the proof at the end. Pure beauty!
All the different properties and proof are amazing to see and learn. I liked all of them.
Most memorable part:
is that as a middle school boy I was obsessed with infinite sums, some of which are in this video,
I never knew how important they were.
Most memorable part was looking at the blue curves inside the 1x1 block and thinking "well sure it's less than one, but is it really negligible? It's still more than a half" and then you asked us to state why it's obvious that it's more than one half.
Most memorable moment? For me, it had to be noting the stack of infinitely many "blue bits" would fit inside the unit square before you mentioned it! As for the proof of gamma > 1/2, each slice of that stack would equal one half if the curved part of the shape were in fact a straight diagonal; since the curve is convex with respect to the "blue bit," the shape's area is greater than half of the rectangle it occupies.
It's really gratifying to be on top of a concept in a Mathologer video, considering that in a typical video (including this one!) there are usually parts that I have to go back and rewatch to make sure I understood correctly! 🤣
"While you often encounter beauty in the optimal, this is definitely not always the case." at 10:01. And then I think of people, personalities. An interesting truth.
This channel is such a joy! Loved the leaning tower of Lira :-)
Never an an integer as the value for any partial sum - that is mind-blowing, since the series runs forever and there are an infinite number of candidate integers along the way, yet the sum misses hitting any integer spot on. It is easier to manage mentally when I recall that there are an infinite number of decimal places demanding all zeroes for a number to call itself an integer.
I really, really, really enjoyed watching this video. Your channel is such a great example for how intriguing and fun mathematics can (and should) be. Even though I can't really pin down my favourite chapter of the video, for me personally, the results by Robert Bailie as well as the fact that the harmonic series' Ramanujan summation is the Euler-Mascheroni constant were the most interesting.
Here in Austria, to get a high school diploma, one must write a ‘Vorwissenschaftliche Arbeit’ (basically a ~25-page thesis about your topic of choice). I wrote about the connection between primes and the Riemann hypothesis and did a lot of research about the harmonic series. I already knew the proof by Oresme, but the rest of the video was mostly completely new to me and I am so thankful for this video.
Liebe Grüße aus Österreich :)
That's great, glad you got so much out of this video :) Alles Gute aus Australien.
You asked which part impresses me most. Well, your enthusiasm impresses me most. I am too old to remember how to prove the divergence of the harmonic series, yet I enjoy everything you mentioned in the video. [Don't bother to send me any of your books. I won't be able to finish reading any of them. :-)
The most memorable part was the leaning tower of lira for two reasons: first, it was a fantastic exercise to have high school students actually perform; second, because it was "coined" by Paul Johnson. Third time I showed this video I realized what was said and started laughing, then had to explain myself.
I liked the part in which the series is approximated to ln(n+1)+gamma. And the fact that the more you go on, the better this approximation is valid. Which, as you said, quite interesting! Usually, the further you go, the worse it gets but not in this case. Very interesting!
EDIT: the demonstration of the no 9s sum is super cool!
Most memorable: finding out the various no 1's, no 2's, no 3's ... no 9's infinite sums are all finite and that you can find an approximate value for the sum of the Harmonic series with a simple formula.
Most memorable part: the connection between the leaning tower problem and the harmonic series. Had already seen an MIT demo of that, but had never attempted to do any calculations
Crazy, how unintuitive things can get
My favorite definition of gamma is the limit as n->infinity of: (1 + 1/2 + 1/3 +1/4 ...+1/n - 1/(n+1) - 1/(n+2)... - 1/n^2 - 1/(n^2+1) ... - 1/(n^2+n) )
I liked most the approximation of the harmonic series with the ln-function, and especially the easy geometric argument, that gamma is between 0.5 and 1.
To have one block beyond the clif line with three blocks, the base needs one with the center mass on the line and the second "floor" two with its center of masses on the extremities of the one on the base
Tristan's visual was really good!
The most memorable part for me was to show that the error between ln(n+1) and the n-th partial sum of the harmonic series was smaller than one. Using it you don't only get a good approximation but also show that the harmonic series doesn't converge.
I'm so happy to see this exercise here. This is an exercise I had when I was in Montpellier Université II Faculty of Science (France) in Mathematics Studies.
And as far as I know, I was the only one to have to idea to start from the top. So the only one I know out of the full amphitheatre) who did the exercise.
Well the rest was not easy we had to tell if it's gonna go infinite or not at the limit. I had to work hard to put it on paper as it was obvious but hard to prove. (For me at least).
But the very interesting fact about this exercise, is, it is THE exercise that made me stop studies. Doing this exercise I dicovered something else, that was of a much higher value.
Watching this video inspired me to make a discovery. You can approximate e^x using the harmonic series. Use the series of integers S[x] defined as follows: start summing the harmonic series until the sum is first >= x. The number of HS terms in the sum is the first term of the series, i.e. S[x](1). Then starting with the next number of the harmonic series, rinse and repeat until you have the number of terms you desire. The ratios S[x](n+1)/S[x](n) form increasingly better rational approximations of e^x as n gets larger. This even works for 0 since S[0] = 1, 1, 1, ... and e^0 = 1/1. And it can be made to work for negative x by using the series for abs(x) and inverting. It can't be the case that I'm the first to notice this, so who is credited with this historically?
For me favourite thing about this video was realizing how the armonic series never lands on an integer. I remember trying some finite sums of the reciprocals with my calculator, when they told us that the complete series was divergent. And now watching this I was amazed by the simple of the argument on how the partial sums always fall outside of the natural numbers.
Hope I win the book!
Explained in a Really attractive manner.
Most memorable part was Tristan's visualisation of the Nein-nein (No nine) thing. After all, I came here for the pretty pictures ;-)
Most memorable: 50/50 tie between the bishops proof and his awful posture
The visual proof at the end was by far the most memorable part
The implication of the cantilever block towers is that an overhang of ANY amount can be achieved given enough blocks. So my immediate questions would be 1) How many blocks are minimally necessary to achieve a particular overhang, 2) What is the maximum number of whole blocks that can be positioned beyond the edge of the cliff for a balanced tower of a) a certain number of blocks, b) a certain height.
My favorite part is the algebraic proof at the end, and that one of the contributors names is “Ooo”!