Let be a sequence of the form
As approaches infinity, the sequence approaches a "continued square root", a number of the following form:
I'll leave it as an exercise to show that if , then this continued square root always converges, and the result is the positive root of the equation .
Let's take as an example. By the formula I just mentioned, we have .
One question I asked myself is, how fast do these continued square roots converge to ?
To put it more precisely, let's consider the sequence , and so on. The decimal approximations for these numbers are 0.5858, 0.1522, 0.03843, 0.0096305, etc. Do you notice the pattern? The numbers seem to be getting about times smaller each time.
Let's make this more rigorous. In the original sequence, we have . Therefore,
So, the differences between the continued square roots and their limit (2) do, in fact, approach a geometric sequence with as the common ratio. Hence, can be asymptotically approximated by a function .
Now, the million dollar question: What is this constant ?
In other words, what is ?
I won't tell you the answer in case you want to solve it on your own. If you succeed, I'd love to see how you solved it so feel free to share it in the comments. If you'd like to see my solution, you can find it here.
Now let's generalize the previous result for the real number . Let's replace with , so that our sequence looks like this: You can show (using a similar proof to mine above) that
Hence, the generalized sequence can be approximated by the function where is the following limit:
Now, I have a conjecture for you. I haven't been able to prove/disprove it yet; see if you can:
Let be a sequence defined recursively by and for , for some . Let . Then,