How to Extend the Sum of Any* Function

In this video, we define a general way to sum a fractional, irrational, or even complex number of terms. Along the way we discover some key concepts from discrete calculus.

This is an extension of the method of my first two videos:
Extending the Harmonic Numbers to the Reals:    • Extending the Harmonic Numbers to the...  
How to Take the Factorial of Any Number:    • How to Take the Factorial of Any Number  


Articles referenced in this video:

Müller, M., & Schleicher, D. (2011). How to Add a Noninteger Number of Terms: From Axioms to New Identities. The American Mathematical Monthly, 118(2), 136–152. https://doi.org/10.4169/amer.math.mon...
https://www.tandfonline.com/doi/abs/1...
arXiv: https://arxiv.org/abs/1001.4695

Müller, M., & Schleicher, D. (2005). How to add a non-integer number of terms, and how to produce unusual infinite summations. Journal of Computational and Applied Mathematics, 178(1–2), 347–360. https://doi.org/10.1016/j.cam.2004.08...
https://www.sciencedirect.com/science...

Müller, M., Schleicher, D. Fractional sums and Euler-like identities. Ramanujan J 21, 123–143 (2010). https://doi.org/10.1007/s11139-009-92...
https://link.springer.com/article/10....
arXiv: https://arxiv.org/abs/math/0502109

The identities at 37:57 can be found on Markus Müller's website:
https://mpmueller.net/fractional-sums/


This video was created with Manim: Community edition: https://www.manim.community/
All the code for this video is available on GitHub: https://github.com/LinesThatConnect/H...


Music credits:
Night Music by Kevin Macleod
river - Calm and Relaxing Piano Music by HarumachiMusic
akebono by Sayuri Hayashi Egnell
Between Moments by Amber Glow
Clockmaker's Daydream by 369
Observations by From Now On
Particle Emission by Silver Maple
Pyrite by Harbours Oceans
The Farmhouse by Silver Maple
Threads by Ever So Blue
The Fog by Lines That Connect
Heavy Head, Light Rain by Lines That Connect
Thanks for Watching by Lines That Connect

The three songs by me can be found on Soundcloud:   / lines-that-connect  


Chapters:
00:00 Intro
1:31 Notes About This Video
2:53 Stating the Problem
4:09 The Recursive Formula
5:52 Going to the Right
7:23 Deriving the Main Formula
10:55 The First Solution
12:57 Flattening Out
14:56 The Forward Difference
16:11 Deriving the Iterated Forward Difference Formula
17:30 The Second Solution
20:18 High Order Forward Differences
23:07 The General Solution
27:01 Nasty Nested Sums
34:04 The General Solution, For Real
36:39 What You Came Here For
37:26 The Published Solution
42:59 Comparing the Solutions
44:58 The Gregory-Newton Formula
45:43 Thanks for Watching!