I am investigating the sequence that tends to the limit $\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \sqrt{a_4+\cdots}}}}$, and although I am making headway on related theory, I would like to graph the sequence, by means of a computer program. (I will define $a_n$, for instance $a_n=e^n$)
The trouble is that the monstrosity is defined 'outside in', as I like to see it.
The issue is that $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$, the limit of $a_n = \sqrt{2 + a_{n - 1}}$, is very easy to compute numerically: one need only instruct the computer to apply the same recursive formula an indefinite number of times -
whereas I have no idea how to translate the move from the nth term to the (n+1)th term of MY sequence into programmable operations.
If there is an easy way to do that, I shall be much obliged. Thank you!
Like continued fractions, if it's not periodic (or some sort of nice pattern), you have to do the innermost first.
See my trial using Mathematica below: