.
By induction, .
Then , so is convergent.
Show that
sqrt(2), sqrt(2*sqrt(2)), sqrt(2*sqrt(2*sqrt(2))), ...
converges, and find the limit.
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So, uh, to show that it converges, we can use the monotone convergence theorem, that says if its bounded and monotone, it will converge... maybe?
EDIT:
Oh, maybe use the theorem that says that subsequences of a convergent sequence converge to the same lim as the orig. sequence? I'm not too sure.
So red_dog pretty much nailed it. I'll break it down for you.
We have the sequence which is strictly increasing and bounded (this would require a rigorous proof...but since you just said show, I'll show it in a simple way).
We have:
See the pattern?
In general,
Use the log rule:
We have: . Divide both sides by and thus: .
Take the limit as n goes to infinity: