nested square roots

What is the value of sqrt(1+sqrt(1+sqrt(1+...)))? Watch this video to find out!

[sciencescience]

Answer is really golden
just like the question .

[nikhilnagaria]

Golden coincidence of Dr Weselcouch and Dr P posting together !

[jimmykitty]

Wow!! It's the golden answer ❤
The golden equation 😍🤗

[renamed]

Why do you replace the part after the first "+" with x? Couldn't you also replace the part after the 2nd, 3rd or basically after any other "+"?
Oh, I just checked and apparently it does not change the result.

[MOHAMMADFAWAZBADEASAIDAL-F-pg]

The answer is equal to the Golden Ratio = 1.6180339

[guill]

Is the nested square root sqrt(1+sqrt(2+sqrt(3+... and so on a transcendental number?

[thomasborgsmidt]

Lovely cello! J.S.Bach? Sounds like that to me!

[mike_the_tutor]

I personally don't like #shorts. Sure, it's somewhat interesting that some complicated expressions can be evaluated in less than a minute, but all these videos feel hectic, or they could be made better by adding a few remarks here and there.

[蔺美云]

I don’t understand the last line. Why it equals empt？

[gdavis]

Why is the golden ratio involved here?

[alvkarthik]

But why replace it with X. It just doesn't feel right. Can anyone one explain.

[pythoncake]

didn't see you prove convergence :P

[kirilrotan]

На самом деле это не доказано. Мы имеем рекуррентное соотношение Xn+1=√(1+Xn), X1=1. Перейти к пределу последовательность мы можем только в том случае если данный предел существует. Следовательно надо доказать существование предела и задача будет решена. Один из способов это сделать применить теорему Веерштрасса, при этом возрастание более-менее очевидно, а ограниченность сверху можно доказать по принципу мат индукции.

[puerulus]

Don't you first need to prove that the limit of the sequence exists and is finite?

[nikhilnagaria]

Wow, but I wonder it should be varphi and not phi.

[mathevengers]

Wow. What a coincidence. My today's shorts is very very similar to this.