The Subtle Reason Taylor Series Work | Smooth vs. Analytic Functions

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Taylor series are an incredibly powerful tool for representing, analyzing, and computing many important mathematical functions like sine, cosine, exponentials, and so on, but in many ways, Taylor series really shouldn't work as well as they do, and there are functions out there that can't be represented with them. What are these functions? And what's so special about so many of our familiar functions that we can compute them with Taylor series?
=Chapters=
0:00 - How to calculate e^x
4:16 - Surfshark ad
5:15 - Why Taylor series shouldn't work
6:54 - A pathological function
8:25 - Taylor's Theorem
10:48 - Analytic functions vs. smooth functions
12:53 - The simplicity of complex functions
14:10 - The uses of non-analytic smooth functions
14:53 - See you next time!
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This video was generously supported in part by these patrons on Patreon:
Marshall Harrison, Michael OConnor, Mfriend.
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CREDITS
The music tracks used in this video are (in order of first appearance): Icelandic Arpeggios, Checkmate, Ascending, Rubix Cube, Orient
The track "Rubix Cube" comes courtesy of Audionautix.com
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The animations in this video were mostly made with a homemade Python library called "Morpho". It's mostly a personal project, but if you want to play with it, you can find it here:
github.com/mor...