How to Encode any Sequence into a Function (Generating Functions)

In the previous video of the miniseries, we looked at the Geometric Series as a gateway to understanding Generating Functions and Taylor Series. In this video, we examine properties of Generating Functions and experiment with them by encoding random sequences and looking for nicer expressions.

Apologies for the finicky screen in some parts of the video!

[jaafars.mahdawi]

indeed very cool! i only wish you collect all these related videos into one playlist for faster access and reference. thank you anyway, keep it up!

[JoseGutierrez-vvuz]

I have an exam tn, you crazy cool fam

[holyshit]

Exponential generating functions can be expanded via differentiation without this extra factor
but not necessarily it is good Idea
E(x, t) = exp(xt)cos(sqrt(1-x^2)t) can be expanded by differentiation using general product rule (also called Leibniz's product rule)
but for E(x, t)=exp(2xt-t^2) better idea seems to be Cauchy product after expressing factor exp(2xt) as a sum of even and odd function
Suppose we have G(x, t) = 1/sqrt(1-2xt+t^2) it is good idea to use binomial expansion twice here but we need to reindex outer sum after using binomial expansion
As a homework think about series expansion of function G(x, t) = 1/(1-t)exp(-xt/(1-t))
G(x, t) denotes ordinary generating function and E(x, t) exponential generating function
Here while expanding x can be fixed

[FFGaming-nfqm]

please make a video on derivation of formula of sum of cubes of n natural numbers that is [n(n+1)/2]^2

[pandavroomvroom]

pretty cool!
what would you do with the generating function of the fibbonacci numbers? i think i remember using it to find closed form for F_n but how exactly do you do it?