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### Sum of consecutive square roots inside a square root

$$\large\sqrt{1+\sqrt{1+2+\sqrt{1+2+3+\sqrt{1+2+3+4+\cdots}}}}$$ I saw this somewhere in the internet but, the website didn't provide me any further information. What is the sum of the equation above? ...
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### Proving $(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}})(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}})=x$

How can I prove this equality? $$\left(\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}\right)\left(\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x+\cdots}}}}\right)=x$$ if $$x>1$$
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### Possible values of infinitely nested square root $n= \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}…}}}$

If $$n= \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}......}}}$$ Is it possible that $n$ is a integer for any $x=Z( \text{zahlen number})$.If yes .What is the value of $x$??
$\lim_{n\to\infty} {\sqrt{1+{\sqrt{2+{\sqrt{\cdots +\sqrt{n}\ }\ }\ }\ }\ \ }\ } = ?$ Either closed answer or an upper bound would help.