(This isn’t my homework.)
Evaluate the limit of this: http://users.adelphia.net/~ybechor/problem.GIF
The solution is:
R = root(2)^(root(2)^(root(2)…
R = root(2)^R
Therefore, R= 2.
Here’s the puzzling part: R can also equal 4. How can this be? A number should only have one limit.
Let’s say what you have is a sequence, and it is defined recursively. We can apply Banach’s theorem about fixed points.
starting point: a(1) = root(2)
recursion: a(i) = root(2) ^ a(i-1)
In order to apply the theorem, we will show the following:
a = 2 is a fixed point (obviously: root(2) ^ 2 = 2)
a(i) will grow towards 2: a(i) > a(i-1) (true, because when we raise x to the power of y, and x>1, y>1 then x^y>x)
a(i) will not grow beyond 2: a(i) < 2 (true, because the power needed to raise root(2) beyond 2 is greater than 2, so once smaller than 2, forever smaller than 2)
That’s it! We proved that the limit of your sequence is 2.
Oh yes: The value 4 is also a fixed point for this recursive definition: root(2) ^ 4 = 4, but you will need a different starting point. With the information given, you may be able to consider different ranges of starting points and their limits.
Sorry, I forgot:
A number has no limit, a number is simply a number. A sequence or a series can have one limit at most (in real numbers). An equation like R = root(2) ^ R on the other hand can have any number of solutions.
Another follow-up:
The first line defines R as the limit of a sequence. The second line defines R as (one of the) solution(s) of an equation. The two lines are related: Any solution from the second line is a good candidate as a solution of the first line, but that’s it. The lines are not equivalent.
