a Pascal identity you probably never heard of

[@ianfowler9340]

Very well presented. Who would have ever thought. Thanks Dr. P.

[@drpeyam]

Thank you so much for the suggestion :)

[@TwoGrids]

e is that one kid in high school who knows literally everyone

[@daniel_77.]

Pi is his older friend

[@Nibor999]

I like this a lot. However, It's not really that surprising if you recognise that (1 + 1/n)^n is simply a binomal.

[@worldnotworld]

How in the world? Pi and e are like church steeples in an old European town that keep unexpectedly revealing themselves when you round a corner. And Pascal's Triangle is rife with churches and innumerable other surprises. Many thanks! (p.s. plus sign at 7:13 was a trivial mistake)

[@juauke]

This is very unexpectedly cool! I didn't think the factorials would all cancel out (I thought we would have needed Stirling's formula)

[@Geenimetsuri]

Love the result! Also liked the enthusiasm! Good vid!

[@alipourzand6499]

Unexpected indeed! Next challenge, pi from pascal triangle ... ☺

[@looksee1001]

Newton did it

[@DDXMalow]

I was just thinking about pascal identity and it’s connection with pascal triangle , thanks for the vid you’re great

[@anwarahmad7076]

Taught about it by Sir Rana in 1999 in our 11th grade in Quaid-i-Azam College, Chaklala, Rawalpindi, Pakistan. 🥰

[@seanoneill2098]

Thank you for sharing this.

[@DiegoGonzalez-zs9wz]

Awesome! What did motivate you to examine that limit? Very fascinating the result!

[@fifiwoof1969]

Grand finale is BEAUTIFUL!!!!

[@codahighland]

Not especially surprising -- given that there's a KZbin video about it, it was either going to be e, pi, or phi, right? And we know that factorials can be rewritten in terms of e using the gamma function. I wonder if you could have done it that way!

[@fifiwoof1969]

9:08 think he says THIS wrong as "future over present and present over past" - if so presents cancel out then it's future over past BUT board shows future over present and past over present - BOARD LOOKS CORRECT THOUGH! Edit: i may have misheard him but again board looks correct - "future over present *divided by* present over past" my initial comment suggests he was multiplying instead of dividing which incorrectly leads to cancellation i mentioned previously.

[@Khashayarissi-ob4yj]

With luck dear Dr, Hoping for more videos.

[@JoeTaxpayer]

This was great. A very unexpected surprise for me as well.

[@doctorb9264]

Where do you find this crazy stuff Peyam !

[@joyboricua3721]

I would, because of the factorial nature of the entries.

[@slavinojunepri7648]

That's a cracking result, indeed!

[@maburwanejohannes4647]

Surprising things happen at infinity. I still can’t wrap my mind around the fact that the sum or product of rational numbers can become irrational at infinity.

[@jesusalej1]

Il capo di tutti li capi!

[@peterhawes9680]

Great video but the thumbnail spoils the surprise 😂

[@pedroalonso7606]

Well, I expected it from the low N value 64/27, but the proof was cool!

[@mab9316]

Oh my God! And now what about the sum of the futur to the sum of the present or to the past??

[@maynardtrendle820]

Very nice!🌞

[@federicosedilis9436]

where did you find this? WOW

[@GoodWill-s8j]

You're awesome!

[@drpeyam]

Thank youuuu!!! 😊

[@GoodWill-s8j]

@@drpeyam 😊

[@jogeshjoshi2535]

Very Excellent 👍👍

[@drpeyam]

Thanks! 😊

[@Michael_Fischer]

My favorit: 0 choose x = sinc(x) = sin(pi * x)/(pi * x) => 1 choose x = sinc(x) + sinc (x - 1) and so on.

[@theproofessayist8441]

How???? How is this possible???? Great discovery!

[@doppler_20]

Why is it (N+1)^(N+1) and not (N+1)^N? Because we are starting at zero?

[@kipu44]

yes

[@godlyradmehr2004]

❤❤❤❤❤ that's really amazing

[@jcfgykjtdk]

Nice 😊

[@drpeyam]

Thanks 😊

[@anamariagonzalezmolina5535]

Fenómeno

[@stlemur]

Everything else is hidden in Pascal's triangle, why not e? :D

[@edilon619]

Gosto de suas camisas.

[@Ali-i9c2y]

Hello from Indonesia Dr

[@BabyXGlitz]

دکتر پیام لطفا از قلم های ضخیم تر با نور بیشتر استفاده کنید، من این قسمت را خیلی دوست داشتم، ممنون

[@vitorbueno6077]

I loved it…

[@drpeyam]

Thank you!!!!

[@robshaw2639]

Where did you find this? I’ve never seen this before

[@drpeyam]

It was recommended by Ian Fowler, a wonderful subscriber of mine 😊

[@DaKeypunchAr]

The only thing that was abnormal to me in the video was him being lefty. But yeah.

[@tylerduncan5908]

The baby is the sum of the parents.... Thats not a family tree i want to be a part of.

[@pamplemoo]

technically kids are the average of their parents 🤓

[@mr.d8747]

*It wasn't surprising because the answer was literally in the thumbnail*

[@xenumi]

🤯

[@daniel_77.]

(Sigh, checks wikipedia)

[@MATH-jh1yb]

OwO! Very very! It's very, very surprising!

[@MohamedElmessoudi-l8e]

Love the content, the erudition and the refreshing, child-like enthusiasm, but why act so effiminate?

[@micknamens8659]

I guess his target audience are children, so he has to entertain them.

[@MohamedElmessoudi-l8e]

@@micknamens8659 Nah. Most of his posts are way above my grasp (and I am an old man), let alone a child's. Recently I did watch another interesting post of his, till the point where he mentioned his 'partner'. That ruined it for me.

[@BabyXGlitz]

I think e should be worshipped

[@antoniocampos9721]

Man, math seems to be funny for you.

[@onradioactivewaves]

I think math drives people to be madmen 😅

[@ВикторПоплевко-е2т]

The notation is discusting

[@gibbogle]

All that remains is to show lim(1 + 1/N)^N = e.