4
$\begingroup$

$\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}$

$\dots\sqrt{2+\sqrt{2+\sqrt{2}}}$

Why they are different?

  • 0
    The second thing must be $\lim_{n\to \infty} x_n$ where $x_0=0$ and $x_{n+1}=\sqrt{x_n+2}$. But expressions like the first one always confuse me a bit. It's obviously supposed to be some positive solution to $\sqrt{2+x}=x$, but it feels a bit ambiguous2011-03-09
  • 1
    A proof of convergence can be found here: http://math.stackexchange.com/questions/11945/limit-of-sqrt7-sqrt7-sqrt7-cdots/11969#11969. With 7 replaced by 2, the proof basically carries over.2011-03-09
  • 0
    @Rasmus Do you know how to rigorously 'define' that first thing? To me it seems both limiting points would satisfy the same equation $x=\sqrt{x+2}$.2011-03-09
  • 0
    @user7992 Answer: They are not, since both numbers equal $2$.2011-03-09
  • 0
    @Myself: I deleted my previous comment because I misread the question.2011-03-09
  • 0
    @user7992: How do you define the first expression?2011-03-09

2 Answers 2