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

Mathologer | 700 years of secrets of the Sum of Sums (paradoxical harmonic series) @Mathologer | Uploaded 3 years ago | Updated 3 hours ago
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_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
smbc-comics.com/comic/math-translations

A very nice list of different sums contained in the harmonic series 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, 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 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: rocketfactorytshirts.com/are-we-there-yet-mens-t-shirt

Enjoy!

Burkard

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$Visualising irrationality with triangular squares NEW (Christmas 2019). Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details) Get ready for some brand new and very pretty visual proofs of the fact that root 2, root 3, root 5 and root 6 are irrational numbers. Root 2 being irrational also translates into the fact that the equation x^2+x^2=y^2 has no solutions in positive integers, root 3 being irrational translates into the fact that the equation x^2+x^2+x^2=y^2 has no solutions in positive integers, etc. What I find very attractive about these proofs is that the destructive core of these proofs by contradiction lead a second secret constructive life, giving birth to infinitely many nearest miss solutions of our impossible equations like for example 15^2+15^2+15^2=26^2-1. Here is the paper by Steven J. Miller and David Montague which features the basic root 3 and pentagonal root 5 choreographies. https://arxiv.org/abs/0909.4913 Footnotes: -our nearest miss solutions like, for example, 15^2+15^2+15^2=26^2-1 correspond to the solutions of the equation y^2 - n x^2 = 1 with n=2, 3, 5 and 6. This is the famous Pells equation, which happens to have solutions for all integers n that are not squares. -there is also a second type of nearest miss solutions like 4^2+4^2+4^2=7^2+1 (a plus instead of a minus at the end). Starting with one of these our choreographies also generate all other such nearest misses. -the original Tennenbaum square choreography and the first puzzle root three choreography generate both types of nearest misses from any nearest miss solution. -The close approximations to the various roots corresponding to our nearest miss solutions are partial fractions of the continued fraction expansion of the roots. -lots more things to be said here but we are getting close to the word limit for descriptions and so I better stop :) Thank you very much to Marty for all his nitpicking of the script for this video and Danil for his ongoing Russian support. Todays t-shirt is the amazing square root t-shirt (google square root tshirt). Note that the tree looks like a square root sign AND that the roots of the tree are really square. Enjoy!$

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