Beyond the Binomial Theorem: The Binomial Series

In this video we will be building up to the Binomial Series. We start with Pascal's Triangle, whose coefficients are found in the expansion for powers of binomials. We then take a combinatorial approach to come up with the Binomial Theorem which applies to positive integer powers of binomials. But what if the exponent is some other real number? Well we can use the power of Taylor Series to come up with the Binomial Series, and along the way we define a natural extension of the "n choose k" notation that work for arbitrary real numbers too.

0:00 Pascal's Triangle
0:51 Expanding Binomials with Maple Calculator
2:00 Connecting Coefficients to Pascal's Triangle
4:03 Combinatorical Approach to Binormial Coefficients
5:12 Binomial Theorem
6:23 Extending "n choose k" formula
8:22 Taylor Series Review
10:00 The Binomial Series
11:22 Binomial Series in Maple Learn

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[posi]

best binomial theorem explain video ever, explain why use c(n, k), relationship with Pascal's triangle. love it

[grinishkin]

4:00 So far, the most valuable note on the topic on Youtube

[angelmendez-rivera]

Here is an addendum: the polynomial α·(α – 1)·(α – 2)·•••·(α – k + 1) actually has a special name: it is called the kth falling factorial of α. This is a kth degree polynomial on α, and the coefficients are known as the Stirling coefficients of the first kind. The binomial coefficient choose(α, k) is this equal to falling(α, k)/k!. This generalization of the binomial coefficient also appears not only as part of the binomial series, but it appears naturally in other contexts. For example, one idea may be to generalize higher-order derivatives to fractional order, and a Newton series, with this generalized binomial coefficient, can be used to explore this idea.

[leapdaniel]

Very cool! Would love to see a video expanding to the concept of multinomials as well.

[PritishMishra]

Your visualisations are getting more and more intuitive to understand!

[geraldsnodd]

0:37 The Sierpenski triangle 🔺️ is hidden inside Pascal's triangle .
When I found out about it I was amazed.

[RealLoveDragon]

Only man who can explain difficult concepts in minutes!!

[MelodiCat]

Extreemly extremly cool generalization. I loved also the way you paired up the binomial expansion to make itobvious why it is n choose k. Wonderful work!

[aashsyed]

Before seeing the whole title I was like this is gonna be multiminomial expansion, and it was not . But do that topic in another video please! The shirt is so cool !!

[jkgan]

I learnt so much in this video it's kinda crazy

[zaydmohammed]

The timing of this video being released was awesome coz just today I was thinking about how newton calculated pi with the binomial expansion of (1+x) ^1/2 nd then realised that the formula for binomial theorem can't take 1/2 as an input so this cleared alot for me.

[wuyqrbt]

And ...
200k .... You almost done.
Congratulations 👏🎉

[TrinityTwo]

The Binomial Theorem, along with the Fundamental Theorem of Algebra, has to be among the most important concepts in intermediate algebra.

[Xanade]

Did... did you just read my mind or something? These are the topics I've been working in four of my classes these weeks. They're advanced pre-SAT(well.. not in US, so sort of) classes, so I'm on binomial expansion and I always add the generalization with this exact notation of alpha choose k. haha

[samsunnahar]

Thanks a lot for excellent explanation!

[eriktempelman]

Cool fact: in the triangle's prime numbered rows, all terms excerpt the 1's are multiplies of that row's prime numbers. E.g. row 5 is 1-5-10-10-5-1, and indeed, 5 and 10 are multiples of 5. Sweet.

[AlessandroZir]

and what happens when I have values for which the Taylor series won't work? when x is not between 1 and -1?!

[Michael_Fischer]

0 choose x = sinc(x) - Replace ! with the Gammafunction and compare with Euler's reflection formula.
1 choose 1/2 = 4/pi

[SuperDeadparrot]

How do you compute Pascal’s Triangle when N is large?

[albinbiju]

I didn't actually get why the series wouldn't work outside the interval -1<=x<=1, why was there the need for this condition? I am a student no mathematician...so plz ignore me if this is a stupid doubt..