Binomial to Multinomial to Multiseries | Algebraic Calculus Two | Wild Egg Maths

We want to move from the familiar Binomial theorem to the more general Multinomial theorem, and then on to less familiar territory involving what we call the Multiseries theorem. But we will have to be rather careful in making this transition. A first step is to think more generally about binomial coefficients to facilitate the step to multinomial coefficients. We will work our way up slowly ---leaving you with a nice challenge to determine the important pattern here!

And along the way, we will have to say some critical things about the modern treatment of "infinite sums", which is of course seriously misguided.

Video Contents (thanks to Lucas Lofaro)

0:00 Intro
0:13 Overview
1:03 A More General View of the Binomial Theorem
2:45 The Trinomial Theorem
5:55 Multinomial Coefficients
9:14 Multinomial Theorem
13:28 Example of Multinomial Expansion
17:38 Towards the Multiseries Theorem
22:18 Problem with Infinite Sums
25:37 Motivation from Newton's Polyseries Theorem
29:36 Multiseries Theorem
33:40 A Preview of the General Pattern

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[sacul]

Video Contents

0:00 Intro
0:13 Overview
1:03 A More General View of the Binomial Theorem
2:45 The Trinomial Theorem
5:55 Multinomial Coefficients
9:14 Multinomial Theorem
13:28 Example of Multinomial Expansion
17:38 Towards the Multiseries Theorem
22:18 Problem with Infinite Sums
25:37 Motivation from Newton's Polyseries Theorem
29:36 Multiseries Theorem
33:40 A Preview of the General Pattern

[bananamanjunior]

Great videos sir.
They have taught me so much, thank you.

[JoelSjogren]

When you define an infinite identity as holding whenever all of its finite truncations hold, this is still a slight departure from algebra and into the realms of logic. An even more algebraic approach could keep track algebraically of the uniform reason that all those finite truncations hold. I am not sure how interesting that would be, but still. One would probably need a vast repository of identities before attempting it.

[SavaConrad]

I do know the pattern for the coefficients, but not for a closed form expression that is simple. I'm used to your clear and careful thinking and I'm excited to see what you come up with.

[davidkeirsey]

Order number of the Monster group squared. c^2. (speed of light squared). 15 supersingular dimensions, 15 degree polynumber. 15 the first odd Abelian group. Planck time.