Mathologer - 2020-11-21
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 https://arxiv.org/abs/0707.0093 Worm on a rubber band paradox: https://en.wikipedia.org/wiki/Ant_on_a_rubber_rope Chapter 3: Proof of divergence Here is a nice collection of different proofs for the divergence of the harmonic series http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf 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 http://www.math.ucsd.edu/~ronspubs/64_07_reciprocals.pdf 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 https://www.smbc-comics.com/comic/math-translations A very nice list of different sums contained in the harmonic series https://en.wikipedia.org/wiki/List_of_sums_of_reciprocals 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, https://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 https://youtu.be/vQE6-PLcGwU?t=4019 I say "at lest ten 9s series". That should be "at most ten 9s series" Today's music (as usual from the free YouTube music library): Morning mandolin (Chris Haugen), Fresh fallen snow (Chris Haugen), Night snow (Asher Fulero), Believer (Silent Partner) Today's t-shirt: https://rocketfactorytshirts.com/are-we-there-yet-mens-t-shirt/ Enjoy! Burkard Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)
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
@Jane Ross t
I lost my life too!
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.
That's it:)
Why is the total area of those triangles 0.5?
@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 ∞
@Kyle Moore , 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.
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.
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.
How often do you imagine an answer different from the answer sheet is actually correct?
@Jetze Schaafsma In math, more often than you think.
In third-year engineering computing, I was fascinated by recursion. I wrote a five line routine to solve a problem that otherwise took pages of code, as my classmate solved it. The graders failed my solution, accusing me of plagiarism, because a student like me couldn't possibly figure that out, although they couldn't come up with anything I might have plagiarized. I got mad and explained how it worked in excruciating detail, so they gave me full marks. I still can't figure out how an automotive differential works. By contrast, another classmate - who started as a car mechanic changing mufflers - understands differentials very well and was frustrated by my incomprehension. He later went into space, but now operates a garlic farm. Life and math are inscrutable. Thanks for listening.
I also had a momentum, when my math teacher could not understand my proof, but said: "I can see your logic, give you the credit."
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
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
Mathologer video series are definitely better than any Netflix series. They surprise me anytime.
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!
For me, the "no integers" part was the most memorable, but honestly the whole video was of great quality (as expected).
Great stuff. It's always amazing how you manage to find such intuitive explanations. Most memorable is probably the "no 9s" visualization.
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
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.
@Kamran Yeah, that was a fun one to just suddenly get (though, like he said, it was “obvious”).
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).
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".
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
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
The weirdest thing you showed is definitely the unusual optimal brick stacking pattern.
@Noobita What if you remove one of those three blocks and instead place it in the fourth row on the left-most block? Wouldn’t this new tower still be stable but with its center of mass a bit more left so that you could shift it a bit more to the right?
@MysteryHendrik maybe it will disbalance the leftmost block on which you are trying to place it. Again this was solved using computer so this must be the optimal solution 😛
@Matthias Reik I agree with you. Those three ones to the top left look suspicious and I have the same feeling. If you placed them in "leaning tower of Lire" way to the left, the whole structure still will be balanced, while giving a little bit "counter mas" to the bricks sticking out of the edge to the right. Proof? Easy: The only brick those three interact with is the one directly under them. So as long as their balance is not broken, the whole structure will stand. Probably we would not gain much with this move, but if this is optimal construction I would expect from it to utilize every possible improvement and those three blocks straight on each do look like can be still improved a bit.
@Adam Markiewicz9 indeed, the only brick those three interact with is the one directly under them. But as you can see, their combined center of mass is already over the leftmost edge of the brick below. So you can't shift their center of mass any further to the left; they'd fall off :)
You could increase their horizontal reach, but not their capacity as a counterweight
@Lieven van Velthoven Interesting, you might be right... Thanks for sharing!
You’re so good at starting simple and yet including stuff that’s interesting for the fully initiated! Great work!
What a wonderful video! Question: the leaning tower of Lyre can be expressed as a harmonic sum, but a physics free body approach to this problem can also show that the way to create an arch with only compressive loads and no tensile loads is to have the blocks follow a catenary curve. Could it be possible then to generate catenary curves using a harmonic sum?
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.
Great video, watched with my husband! My favourite part was the visualization of the no 9s series, thank you Tristan for the idea!
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. 🤣
@Stephen Mc Ateer That was exactly my same train of thoughts. :D I keep jumping between "no way!" and "ah, but it's obvious".
@Jack W You can make that arbitrarily probable too ... just need MORE DIGITS!!! :D
@Davide Aversa you
Are right. It seems implausible and obvious at the same time depending on how you look at it. So much fun.
@Stephen Mc Ateer No. You are not requiring at least 100 zeroes, that series would diverge. You are requiring EXACTLY 100 zeroes, which means you can't have any more than 100 zeroes, which is what makes it converge.
Around some enormous number X with L=log_10(X) digits, the total number of numbers is X=10^L, while the total number with exactly 100 zeros is (L choose 100) times 9^(L-100). The fraction of numbers with exactly 100 zeroes is (9/10)^(L-100) times approximately L^100 times (10^100/100!). The integral approximation (integrating over L) is (integral) L^100 (0.9)^(L) from 100 to infinity dL which would be 100! if it were a pure gamma function, so the combinatorial factors mostly cancel. This approximation is good enough to get estimates, I presume the fellow did similar integrals in the paper.
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!
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!
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.
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.
-----------------------------------------------
Thanks
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.
Would be interesting what your kids would make of the animation of this proof :)
@Mathologer Maybe I'll use it in my "Mathe AG". :)
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.
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.
I liked the crazy optimal overhang tower the most. Didn’t expect that at all.
Wow this is so fascinating! Loved learning about all these amazing properties!
The fractal pattern is amazing, but it makes me wonder how does it relates to the reciprocals geometrically
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
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.
24:13 is it already known that you can do a similar thing with the positive integers series? If you graph the partial sums formula of the positive integer series(triangle number formula), then graph y=x, the area between the two intersections of the graph has an area of 1/12.
Most memorable part: the visualization of the "no nines sum convergence"
What an awesome way to look at it.
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.
Most memorable: that the harmonic series narrowly misses all integers by ever shrinking margins
@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.
@Dave Langers 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!
@Parker Shaw Smart!
It even misses all real, complex and otherwise numbers except its own partial sums. But that somehow doesn't have quite the same ring to it... ;-)
crazy stuff
This was amazing! Results which seemed completely inaccessible at first glance made simple to see. My most memorable part was that crazy formula you flashed on screen involving e, gamma and the prime numbers.
The most memorable part is the connection of the 'γ' and the log() function to the harmonic series! Really amazing!!
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!!!
Most amazing part for me was the addition of constant gamma.... I get that the extra area would be greater than 0.5 but involving gamma in the approximation, dramatically changes the situation ! It is so cool. I love it :)
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.
In my ten years or so of watching YouTube, this is the video that made me feel like I have to comment.
My favorite was learning the 100 zeros series was finite. I thought in my mind, that series is more than the harmonic series multiplied by 1/Googol, and if the harmonic series is infinite, the 100 zeroes series should be too. I know I’m probably wrong, would love to figure out why!
Anyway, more importantly - Thank you for your amazing videos, your energy, and your excitement for maths. It’s been a pleasure seeing these videos over the years and really getting to understand some fairly complex math without any academic background in it. :)
Awesome video! Your excitement is truly contagious through the video :D
Burkard thank you for you wonderfully accessible, stimulating, and enlightening videos. My favorite aspect was how the parabolic and customer staged tiling provide better solutions then the greed algorithm. I also loved your "Are We There Yet" t-shirt. I don't see it on your store. Do you have a pointer to where it can be purchased?
The crazy maximum overhang tower was the most memorable for me. There's something really mind-blowing about generating a complex structure from a relatively simple problem, a bit like how the Mandelbrot set is defined.
gamma is greater than 0.5 because each part above the function is a little bit more than a right angled triangle, all of which fit into a square 1×1, so their total area is less than 1/2(1×1) which is 0.5 🙂
Nice, I was thinking the same thing, so it's good to get confirmation that I was on the right track.
I thought of the same thing but in terms of rectangles being covered more than half :)
Saw that one too ... made my day!
you beat me to it :D, although you can get a more rigorous proof of this as well.
You know that 1/x is strictly decreasing and also convex (its derivative 1/2x^3 is positive on (0,∞)), by the definition of convexity it lets you prove that for y ∈(n,n+1) implies that 1/y<g(y), where g(y) is the line passing through the points (1/n,n) and (1/n,n+1), so as a result, this lets us conclude that
∫1/x < ∫g(x)
So our goal from here is to prove that for the harmonic series h(x) because gamma is defined to be
γ=∫h(x) - ∫1/x.
Then if ∫h(x)-∫g(x)=.5 then γ>.5. Well think about the 2 integrals as ranging from a natural number n to n+1, then we can evaluate both ∫h(x) and ∫g(x) on that interval like so
∫h(x)-∫g(x)=1/n*(n-(n-1))-1/2(1/n+1/(n+1))(n-(n-1))
∫h(x)-∫g(x)=1/n-1/2(1/n+1/(n+1))
∫h(x)-∫g(x)=1/2(1/n-1/(n+1))
From here you have a pretty simple telescoping series that you can simplify
1/2((1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...)
So the end result is that ∫h(x)-∫g(x)
@Adityan Singla Yeah, and the curve always drops below the straight diagonal so the part of the rectangle above the curve is always bigger than half.
The most memorable part for me, as a math & comp sci student, was learning about Baillie's paper! It promises to be a fun weekend read haha
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!
Most memorable: Tristin’s visual proof of the finiteness of the no-9 series.
Hmm, I love the simplicity of the no 9s proof, so I'm going with that. Figured it'd be connected to a geometric series. Very nice!
Daemos Magen - 2020-11-22
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.
Zakir Reshi - 2021-11-13
🤣🤣🤣
Virgocygni56 - 2022-01-19
THESE WIFES DO NOT UNDERSTAND US....:-(
pekka puupaa - 2022-02-01
ä
alainmcin - 2022-02-06
LOLLL
peteiis easy - 2022-04-05
@Douglas Strother 💀