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

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

Chapter 3: Proof of divergence

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

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

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

If you still know how to read :) I recommend you read the very good book Gamma by Julian Havil.

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)

Enjoy!

Burkard

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[daemos_magen]

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.

[bernyelpro]

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

[sergeboisse]

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

[shivambiswas]

Any divergent series: *exists*

Ramanujan: Allow me to make it convergent.

[Fun_maths]

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

[mattbox]

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.

[jzieba]

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

[_kapy_]

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

[apbmes]

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.

[anon]

They should have called it the 'Barely Divergent Series'

[theknger]

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. 🤣

[nnamgiven]

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!

[zacharystark]

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

[whatby]

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.

[davidgustavsson]

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

[johnchessant]

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.

[dragifire]

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!

[peter_p_r_zhang]

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.

[karateoscar]

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

[dennisbell]

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