Complex Analysis 9 | Power Series

[@leonilsonnunes3755]

I always stay anxious for next video of this channel. These videos are true Math classes.

[@hyperduality2838]

Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda.

[@rob5467]

It would be great to see the details of the proof you reference in the video

[@brightsideofmaths]

I put it in my book about Complex Analysis :)

[@thermrm]

Thank you so much for making these videos. I'm a physics student, and these concepts aren't taught this well. Also, your handwriting is so neat! love from Palestine

[@hyperduality2838]

Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda.

[@jesusjaviervidalesperez5238]

i need the proof of that elegant fact

[@JojiThomas7431]

Good videos. Really understood the concept

[@hyperduality2838]

Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda.

[@NewDeal1917]

00:00 Intro 0:57 Why power series are important? Example of exp(z) 2:05 General definition for C 4:03 Example. Geometric series + conditions for convergence 9:06 Cauchy-Hadamard theorem

[@hyperduality2838]

Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda.

[@TrongNguyen-sz4wr]

thanks very much, teacher. Its really help me a lot :)

[@brightsideofmaths]

You are really welcome. Please also check the pdf version of this video :)

[@darrenpeck156]

I would very much like to see the proof.

[@manswind3417]

Hey! Fantastic video, really well-made and comprehensive, and easy to understand as well. I have one request tho, if possible please include proofs of the theorems you mention (such as Cauchy Hadamard, or that power series converge in a ball), or at least provide some links for the same. Much appreciated, thanks!

[@brightsideofmaths]

Thank you very much. Most stuff about power series is proved in my real analysis course: tbsom.de/s/ra

[@brightsideofmaths]

I am also writing a book for this course, which supporters can already download. There, I fill in all the details.

[@AJ-et3vf]

Great video! Thank you!

[@khalidmohammadjama]

Another nice job 👍

[@omkardwivedi513]

HEY NICE JOB . I AM YOUR GREAT ADMIRER. PLEASE MAKE VIDEOS FOR GEOMETRY FROM (EUCLID TO COORDINATE TO MODERN ONE)

[@darthtleilaxu4021]

Power Series...?! Haaaa ! "Séries entières" ! Thanks for the video!

[@mariolemelin27]

As always, great videos. Could you give us a link here to the proof of the nice result (8:48) as a pdf. Thank you in advance!

[@brightsideofmaths]

Good idea. I have my real analysis book where you find a very similar result. But I can also prepare a video or pdf here. It will take some time.

[@mariolemelin27]

@@brightsideofmaths A pdf would be enough I think! That would be already great.

[@brightsideofmaths]

The proof is now written down in the book about this course. The book is not finished yet, but supporters can already download the preview.

[@higifnr]

I have a friend that is studying abroad in Germany and when he showed me the Cauchy-Hadamard limit I was kinda confused because we use the d'Alembert rule and it seems to be a lot simpler. Any reason why to use C-H over it?

[@brightsideofmaths]

The root criterion is more general than the quotient criterion, which you mean I suppose?

[@higifnr]

@@brightsideofmaths Yeah I meant lim |an+1/an| = 1/R Can you please give me an example of how this criterion is limited? 😊

[@brightsideofmaths]

You cannot apply it immediately if some sequence members are zero, can you?

[@masteroogway7624]

Does anyone know how he makes these presentation?

[@brightsideofmaths]

I know it!

[@masteroogway7624]

@@brightsideofmaths willing to share? XD

[@brightsideofmaths]

@@masteroogway7624 Sure. I use Xournal :)

[@masteroogway7624]

@@brightsideofmaths thank you very much! I am an aspiring teacher and your videos have helped me organise so much. If you have time and are intested, I would love to see your work process for these videos, they are incredible!

[@hyperduality2838]

@@brightsideofmaths Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda. Energy is duality, duality is energy Everything in physics is made from energy including mathematics!

[@hyperduality2838]

Convergent is dual to divergent -- power series are dual! Syntropy is dual to increasing entropy --the 4th law of thermodynamics! Analysis or wholes become parts is an entropic process or reductionist process -- divergent. Synthesis or parts become wholes is a syntropic process, or a holistic process -- convergent. Noumenal (analytic, rational) is dual to phenomenal (synthetic, empirical) -- Immanuel Kant. "Always two there are" -- Yoda.

[@yomo_13]

Plz make the prove

[@brightsideofmaths]

It's in my book, which you can download for free as a supporter.

[@ayaayout1202]

Hello May i have your email ? I would like to ask you few questions .

[@brightsideofmaths]

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