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

From Mathologer.

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

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/