let
and
Then
If the limiting value of is , then the limiting value of is also .
So
So we see that
Note that the given sequence is increasing and bounded above (you can show easily by induction on n that each member of the sequence is bounded by 2). Thus the sequence is guaranteed to converge.
Soroban - I found a little mistake in each of our arguments. The answer is actually 4.
In my argument, I misread the problem and left off the constant 2 in front. Since constants can be pulled outside limits this is no big deal, and you can just multiply my answer of 2 by 2 to get 4.
In your argument, the right hand side of where you wrote "Square" should actually start with a 4 (since 2^2 = 4).
Here is my corrected solution:
Let
and
Then
If the limiting value of is , then the limiting value of is also .
So
So we see that
Note that the given sequence is increasing and bounded above (you can show easily by induction on n that each member of the sequence is bounded by 4). Thus the sequence is guaranteed to converge.