TITLE: Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .
QUESTION [5 upvotes]: Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ?
How to find if it is convergent or not?
Thanks!

REPLY [7 votes]: Hints:
$$z=\sqrt{i+\sqrt{i+\ldots}}=\sqrt{i+z}\implies z^2-z-i=0$$
and you're left with a simple complex quadratic to solve
Note: Formally , you first should prove the sequence $\,\{\sqrt i\,,\,\sqrt{i+\sqrt i}\,,\ldots\}\;$ has a limit. More formally even, one should also choose a branch for the square root...but you can "wave" these in a first approach.