I have been using this identity for exponential function for so long... It's nice to finally see a proof! One of my favorite application of this property was in statistical mechanics where we would express some systems with the grand-canonical potential to later derive the thermodynamic properties of those ensembles. We would define a grand potential function that had a sum of the microstates within an exponential function and it was often useful (or cute) to represent this with the Pi notation (product operator).
This explanation is so good ! Thank you so much for the explanation really :,( your channel is underrated.. I hope you get more audciance you deserve it. Thank you again for the clean explanation :D !
Was going through Penrose's Road to Reality and he had this problem of proving from the power series definition that exp(1/2)^2=e. Only hint was the definition of binomial coefficient. One long rabbit hole later and this video was a perfect explanation. Thanks!
Love the way you explain things and the clear presentation in your videos. I have seen several other sources state that both a and b must be absolutely convergent for the Cauchy product to be absolutely convergent whereas you suggest that one of a and b may be conditionally convergent. Can you clear this up please. Thanks.
![Real Analysis 22 | Cauchy Product [dark version]](/img/n.gif)
Your table reminds me of the way Cantor laid-out the Rational Numbers to show that they are countable.
Example is just absolute beauty !!...thanks
I have been using this identity for exponential function for so long... It's nice to finally see a proof!
One of my favorite application of this property was in statistical mechanics where we would express some systems with the grand-canonical potential to later derive the thermodynamic properties of those ensembles. We would define a grand potential function that had a sum of the microstates within an exponential function and it was often useful (or cute) to represent this with the Pi notation (product operator).
This explanation is so good ! Thank you so much for the explanation really :,( your channel is underrated.. I hope you get more audciance you deserve it. Thank you again for the clean explanation :D !
Glad it helped!
Was going through Penrose's Road to Reality and he had this problem of proving from the power series definition that exp(1/2)^2=e. Only hint was the definition of binomial coefficient. One long rabbit hole later and this video was a perfect explanation. Thanks!
Thank you very much :)
so in a way cauchy product formula transforms a product of series into a *composition* of series?
So foiling is distributive at 1:25?
It's a finite sum or what do you mean?
Love the way you explain things and the clear presentation in your videos.
I have seen several other sources state that both a and b must be absolutely convergent for the Cauchy product to be absolutely convergent whereas you suggest that one of a and b may be conditionally convergent.
Can you clear this up please. Thanks.
Wow, thank you! If you look at the proof of the statement, you will see that we only this assumption.
You can find my lecture notes about the topic to read the proof :)
Beautifully done!
Can somone explain to me why every "package" only has finitely many elements ?
Do you see the picture in the thumbnail? That explains it already. Each package is a finite ellipse/package there.
how did you get the formula for c_k in the example?
You just put in the elements with the correct index :)
love u
Soo beautiful