>>14036666Well let's define a sequence as
and suppose, for now, that .
Of course converges it will go to the golden ratio as some people have pointed out (the positive one).
However lets see what happens for different values of .
If : which can be easily proved it doesn't converge. (very similar to )
Notice that after the first iteration the sign of the starting value won't matter so .
If : It can be easily proved that the series is strictly monotonic.
If : , so again it is strictly monotonic.
If : It can be proved that . (so it can't converge to the golden ratio)
If : so we're in the previous case.
If to be honest I can't track what's happening in this general case however by this point it is clear that if we let then so it will infact converge to the golden ratio (it will be ). Now I have a feeling that it won't "balance" for other values but I can't think of a way to prove it. I would love for someone to give it a try.
