I have come across a few different questions relating to my issue (namely this one, but the answers are not working for me. Here are my inputs;
{Slider[Dynamic[c], {0, 2}], Dynamic[c]}
Dynamic[RecurrenceTable[{a[n + 1] == (c + a[n]^2)/2, a[1] == c/2},a, {n, 1, 20}]]
Dynamic[ListPlot[RecurrenceTable[{a[n + 1] == (c + a[n]^2)/2, a[1] == c/2},a, {n, 1, 20}]]]
These work so far and are just to help me visualize the sequence and its values. Then I try
Dynamic[RSolveValue[{a[n + 1] == (c + a[n]^2)/2, a[1] == c/2}, a[Infinity], n]]
(* RSolveValue[{a[1+n]==1/2 (0.715 +a[n]^2),a[1]==0.3575},a[\[Infinity]],n] *)
And I can't seem to get it to output the limit (when one exists, namely when c is less than or equal to 1). I've tried a couple of the different solutions from that thread but I can't get it to work. Most of the time it just reprints my input like the above. I'm trying to avoid defining my function because then I have to shift the index (n+1 to n), which I don't want to make a habit of.
I'm new to Mathematica so my apologies for any mistakes. Is there a way to get this thing to give me the limit or let me know if it doesn't exist?
In[919]:= Solve[ainf == (c + ainf^2)/2, ainf] Out[919]= {{ainf -> 1 - Sqrt[1 - c]}, {ainf -> 1 + Sqrt[1 - c]}}
. It is not difficult to reason by induction that the first is the correct limit:c<=1
by assumption, soa[1]<=1/2
and for alln
,a[n+1]<=1/2+a[n]^2/2<1/2+1/2
. $\endgroup$-1<=c<=1
. For values wherec<-1
some experimenting indicates we have limit cycles. $\endgroup$