Extending the Harmonic Numbers to the Reals

The harmonic numbers are the partial sums of the harmonic series - sums of whole number reciprocals. This video explores how we can extend their domain to the entire real line.

This is my entry for the Summer of Math Exposition 1.

00:00 - Intro
1:45 - Graphing the Harmonic Numbers
2:47 - A Recursive Formula
4:23 - Using the Recursive Formula
7:33 - The Super Recursive Formula
8:52 - Finding the Interval
11:27 - Example: H(0.5)
11:59 - Deriving the Solution
13:10 - Graphing the Solution

#SoME1

[LinesThatConnect]

Thanks for watching everyone!
I obviously hoped to see a positive response, but I didn't think I'd actually get thousands of views on my first video! It makes me really happy to see so many people enjoying this topic. The Harmonic Numbers are the tip of an iceberg into some really cool math, and I hope to eventually share that whole journey on this channel.

I've been working on another (completely unrelated) video, but as I'm doing this between work and other personal projects, I can't give any estimates as to when it will be ready. I just wanted you to know that there definitely is more on the horizon!

[sebastiangudino]

Guys, get ready, we are literally witnessing the birth of a new legend in the math-educational YouTube scene. This is going to be great, i can feel it

[eammonful]

This was excellent. I've watched a lot of the SOME1 videos and this is easily one of the best. Do you expect to release any more anytime soon?

[minyiiiii]

6:00 that pun was so good! sneaky, unobtrusive, and perfect in context. subscribed.

[FareSkwareGamesFSG]

When I first learned about sigma notation and the Riemann Zeta function, I spent the next 5 nights playing on Wolfram Alpha, and it was fun. It was very similar to what you've done here! Thanks for the connection with the digamma function, too, I never knew what that was.

[DrZye]

Great visualization, great pacing, interesting topic!

[鸞麤]

Nice video. One very small point I would make would be at 10:55 on your justification of using a straight line for the interval, as opposed to some squiggly up and down one. I believe it would be better to justify it under the intuition that the harmonic functions extended to the reals should be a strictly increasing function, rather than the rather loose "it is the most natural. It's probably being overly pedantic, but mathematics is all about rigour

[hemanthkotagiri]

I didn't even notice 15 minutes have gone by. That's how good you are at explaining things. Awesome, man! Keep going! The world needs more lucid explainers like you!

[jackrookstool]

Dude, this video is one of the best produced math videos I’ve seen in a while! You have the elegant animations of 3Blue1Brown while also touching subjects I’m more interested in, so bravo, honestly! Keep up the good work, I’m excited to see what you have in store in the future!

[ЧингизНабиев-эг]

Damn, I am a high schooler who is interested in maths. Gamma-function, Digamma function, harmonic numbers and extension of series from integers to real(and complex) numbers are definitely one of my favorite topics. Honestly, this is the best video about harmonic numbers I’ve seen so far.

[zanzlanz]

Amazing first video - I can't wait to see what's next!

Not taking into account the reasons why you'd prefer one notation over another, I think it's a bit curious though how computing the first numbers gets more and more difficult as you get further into this video - I mean at the start it is just evaluating fractions, and by the end you have to calculate an infinite sum!

[avz]

Wow! I used essentially this idea to find a continuous extension for the factorials! I know the gamma function already exists, but doing it this way gives you a different formula that happens to give you exactly the same thing as the gamma function. I never considered doing the same thing with other functions, so cool!

[telnobynoyator_]

I am so glad for 3b1b's Summer of Math Exposition, great videos are popping up everywhere !

[andriypredmyrskyy]

I discovered this while "discovering" the stamp collector's problem for myself. There's a fun approximation that's useful if you're trying to find how long it takes to get items from a random draw.

[TagetesAlkesta]

I plugged the formula into Desmos on my phone and it crashed lol

[benjaschunk]

Next step: generalize it to complex numbers

[ERROR-eiyv]

As someone who's interested in somewhat niche generalizations like this, this video was really interesting! It was very well explained and visualized and easy to understand

[mahxylim]

12:28 Math is crazy in a way that you can have an good intuition about every separate fact but when combined, knocks you back to the start and realize you don't really comprehend the whole story.

[piyushmahamuni]

I can see that you took great effort in making sure that every step and intention behind it is clearly conveyed. When you're an expert yourself, it's hard to know what steps would be difficult to digest from a novice point of view. I enjoyed watching this video very much.

[danidotexe_]

Got excited that I'd found a new cool maths channel. Got disappointed when I realized it's only your first video. Got excited again when I realized it's only your first video. Please keep it up!