You could reach a compromise of
k
k+1
k+1/k
k+1/(k+1)
k+1/(k+1/k)
k+1/(k+1/(k+1))
...
In this way, the values for k=-1 become:
-1,0,-2,[\tex]und,-1.5,\to-1,-1.\overbar{6},\to-2,-1.6,\to-1.5,-1.625,\to-1.\overbar{6},..., tending towards -\frac{1+\sqrt5}{2}
And for k=0...