Inductive factorial sum visual proof

This is a short, animated visual proof demonstrating a finite sum involving products of factorials. The proof exploits the classic inductive proof of the formula in question.

#mathvideo​ #math​ #numbertheory #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #factorial #squares #factoradic #discretemath #inductionproof #induction #factorialnumbersystem #finitesums #sum #series

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[_____]

Math is beautiful. Intuitive and logical.

[SpeezyBeez]

Gives me the good old vibes of studying the night before the exams!
a true memory reboot this is

[charliezard]

That’s cool. Does that help us do or understand anything better though?

[aryamangupta]

If we consider this as a partial sum, how do we check the convergence of the series?

[DevTech]

n n!
(n+1-1)n!
(n+1)n! -n!
(n+1)! - n!
Put values of n
2! - 1!
3!- 2!
...
....

(n+1)! - n!

(n+1)! - 1

[deeneditor]

or you can do what's called "une somme télescopique" (sum(1 to n) a_k+1 - a_k = a_n+1 - a_1 )

sum(1 to n) k*k!
=sum(1 to n) (k+1-1)*k!
=sum(1 to n) (k+1)*k! - k!
=sum(1 to n) (k+1)! - k!
= (n+1)! - 1!
= (n+1)! - 1