- #1
lokisapocalypse
- 32
- 0
Hey guys,
I have a sequence, [tex]\sqrt{2}[/tex], [tex]\sqrt{2 \sqrt{2}}[/tex], [tex]\sqrt{2 \sqrt{2 \sqrt{2}}}[/tex], ...
Basically, the sequence is defined as x1 = root 2
x(n+1) = root (2 * xn).
I need to show that this sequence converges and find the limit.
I proved by induction that this sequence increases. Since it increases, its bounded below by root 2. I need to show that it is bounded above by 2. Then I can use the Monotone Convergence Theorem to show that this sequence converges.
Any ideas?
I have a sequence, [tex]\sqrt{2}[/tex], [tex]\sqrt{2 \sqrt{2}}[/tex], [tex]\sqrt{2 \sqrt{2 \sqrt{2}}}[/tex], ...
Basically, the sequence is defined as x1 = root 2
x(n+1) = root (2 * xn).
I need to show that this sequence converges and find the limit.
I proved by induction that this sequence increases. Since it increases, its bounded below by root 2. I need to show that it is bounded above by 2. Then I can use the Monotone Convergence Theorem to show that this sequence converges.
Any ideas?
Last edited: