How to Extend the Sum of Any Function скачать в хорошем качестве

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!