Q:

limit of a radical

I want to find the limit of infinite nested radical, but I don't know how to express this in mathematica. I used Fold[] but I couldn't make it exactly like this. 

A:

Will this function work?
ClearAll[f];
f[x_, y_, n_ /; (j = n; True)] := Sqrt[x + f[y, n - 1]] ;
f[y_, n_] := Sqrt[y + (j - n + 1) f[y + 1, n - 1]];
f[y_, 0] := Sqrt[y];

f[3, 9, 5]

A:

A little different formulation, I wonder if its valid to choose any particular value for the last value where we choose to truncate the expression:
val[n_Integer, r_] := Module[{f},
  f[n] = r;
  f[i_] := Sqrt[(7 + i) +  i f[i + 1]];
  f[2]]
val[6, r]

so we can see this converges to the same value  with large n regardless of the terminal value r.
Show[Table[
  ListPlot[Table[{n, Sqrt[3 + val[n, r]]}, {n, 2, 20}], 
   Joined -> True] ,
  {r, {0, 1, 2, 5, 20, 100, 1000}}], PlotRange -> All]

I don't think qualifies as proof of convergence, but its pretty convincing.

