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. The animations for this video were made with the community edition of Manim (https://www.manim.community). Huge thanks to everyone who worked on the library, as well as the members of the Discord server who answered my many questions. This is my entry for the Summer of Math Exposition 1. All the music in this video is available on my SoundCloud: https://soundcloud.com/lines-that-connect 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