Fourier Series From a Divergent Series

[@robmarks6800]

This is amazing!! Please do more of this topic regarding fourier series👍

[@bigcatsworldadaptedpintere7277]

How it is possible? abs(x) is less than one and while abs(e^ix)=1 please exlplain .mercey

[@DavidFosterZen]

Out of curiousity, where did you find the nth derivative of the power series of e^i*theta, to be a "solution" to the Zeta Function?

[@Inversed]

Very cool, I was thinking about what'd happen if we were to sum a divergent Fourier series using something like Cesaro summation and then found your video. One nitpick: you should've used Lancosz sigma factors to combat the annoying Gibbs effect.

[@DavidFosterZen]

A little heresy is good for the soul. Honestly, I am not convinced you were doing bad manipulations with those divergent series. In any case, divergent series are most fascinating topic, and I don't think they are fully appreciated. Most of the time, I just see people trying to explain them away. It is much more fun to take them seriously and see what we can do with them.

[@Math_oma]

I agree that one should occasionally be bold and heretical. It seems like we lose a lot of information about these divergent series by lumping them all into one group, which is really defined by the property "not converging". However, there are several pieces of evidence that there's an internal logic to these series which produces completely reasonable results. I think the Fourier series appearing from a divergent series is just one of many examples.

[@DavidFosterZen]

Ok, you have made me get off my butt and make a reply video. This one is on continued fractions, but I am going to work up to divergent series in later videos. kzbin.info/www/bejne/l6eannuahc-Lmqs

[@johnstuder847]

Thanks so much for your clear explanations. Love how you don't skip steps. Have you seen Stanfords Fourier Transform class by Osgood? I think you could do it way better, in far less time. Would also get you lots more viewers since there are so many applications for Fourier transforms. You said you like to explain convoluted things...well, Fourier transforms invoke lots of convolution! (A*B). - (Also check out Cohl Furey's Quaternions.)

[@Math_oma]

+John Studer Yes, I've worked through most of that course and it's the best course I've seen on Fourier analysis thus far.